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<span class="kw">let </span><font color="Maroon" title="c1">Y</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">X</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">Y</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">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="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> <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="b3">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) &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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Olive" title="b1">X</font></span>)</span>  ex <font color="Olive" title="b5">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u</font> &amp; (  for <font color="Olive" title="b6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> holds <br/> ex <font color="Olive" title="b7">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b8">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b7">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b7">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b6">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">X</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> &amp; 1 <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="Olive" title="b1">X</font> holds <br/> 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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">Y</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Olive" title="b1">X</font></span>)</span>  ex <font color="Olive" title="b5">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u</font> &amp; (  for <font color="Olive" title="b6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> holds <br/> ex <font color="Olive" title="b7">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b8">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b7">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b7">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b6">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) );<br/>
<a NAME="E1:54"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E2:54"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 for <font color="Olive" title="b1">X</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>  st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> &amp; 1 <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="Olive" title="b1">X</font> holds <br/> 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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">Y</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Olive" title="b1">X</font></span>)</span>  ex <font color="Olive" title="b5">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u</font> &amp; (  for <font color="Olive" title="b6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> holds <br/> ex <font color="Olive" title="b7">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b8">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b7">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b7">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b6">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">X</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"> ( <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; 1 <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">X</font> implies  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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<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">X</font>,<font color="Maroon" title="c1">Y</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b4">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b6">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b5">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:54_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
( <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; 1 <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">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">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<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">X</font>,<font color="Maroon" title="c1">Y</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b4">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b6">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b5">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">K</font> = <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">n</font>;<br/>
<a NAME="E3:54_1"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="finseq_3.html#T55" onmouseover="rs('finseq_3/T55')" onmouseout="rh()">FINSEQ_3:55</a></span>;<br/>
<a NAME="E4:54_1"/><span class="lab"><font color="Green" title="E42">A6</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">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="finseq_3.html#T53" onmouseover="rs('finseq_3/T53')" onmouseout="rh()">FINSEQ_3:53</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><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="E1:54_1_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_1_1_1"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<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">X</font>,<font color="Maroon" title="c1">Y</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b4">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b6">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b5">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )</span><br/><div class="add"><a NAME="E2:54_1_1_1"/>
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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>;<br/><a NAME="E3:54_1_1_1"/><span class="kw">then </span>
1 <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">X</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/><a NAME="E4:54_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">X1</font> = <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D10" onmouseover="rs('prvect_2/D10')" onmouseout="rh()">PRVECT_2:def 10</a></span>;<br/><a NAME="E6:54_1_1_1"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c5">X1</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="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:54_1_1_1"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  <a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">V</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a> ex <font color="Olive" title="b3">k</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">X</font> st <br/>( <font color="Olive" title="b2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Olive" title="b3">k</font></span>)</span>,<font color="Olive" title="b2">V</font>) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#D3" target="_self" onmouseover="rs('lopban14/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">V</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>, <font color="Maroon" title="c8">k</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">X</font><span class="kw"> such that </span><br/><a NAME="E10:54_1_1_1"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
( <font color="Maroon" title="c7">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> &amp; <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c7">V</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E8:54_1_1_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">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="c5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c5">X1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">Y</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E12:54_1_1_1"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
( <font color="Maroon" title="c9">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c9">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c9">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="c5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span><br/>  for <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">X1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">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"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b2">x1</font> ) &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="c5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> holds  <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="c9">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> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T38" target="_self" onmouseover="rs('lopban14/T38')" onmouseout="rh()">Th39</a></span>;<br/><a NAME="E13:54_1_1_1"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
 <a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<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="c5">X1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c5">X1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c1">Y</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T25" target="_self" onmouseover="rs('lopban14/T25')" onmouseout="rh()">Th26</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">I</font> = <font color="Maroon" title="c9">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<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">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E6:54_1_1_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E8:54_1_1_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E10:54_1_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E13:54_1_1_1"><span class="lab"><font color="Green" title="E48">A13</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c10">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="c10">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c10">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c10">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b3">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b5">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E15:54_1_1_1"/>
( <font color="Maroon" title="c10">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c10">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c10">I</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E6:54_1_1_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E8:54_1_1_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E10:54_1_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E12:54_1_1_1"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="txt" href="#E13:54_1_1_1"><span class="lab"><font color="Green" title="E48">A13</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">  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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b3">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b5">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E16:54_1_1_1"/>
 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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b3">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b5">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b4">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b3">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b4">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b3">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b3">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">x</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">X</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">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b3">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b4">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b3">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b3">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b2">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c13">g</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:54_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font>
 <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>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</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"> ( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:54_1_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</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">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:54_1_1_1_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</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">  ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">XX1</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <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="c15">h</font> = <font color="Maroon" title="c11">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c14">XX1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">XX1</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">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c14">XX1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Maroon" title="c14">XX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b1">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c15">h</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">XX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c15">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E6:54_1_1_1_1"/>
( <font color="Maroon" title="c14">XX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c15">h</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="c11">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c16">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">X1</font><span class="kw"> such that </span><br/><a NAME="E8:54_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
<font color="Maroon" title="c12">x</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="c16">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="#E6:54_1_1_1"><span class="lab"><font color="Green" title="E44">A8</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">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c12">x</font> = 
<font color="Maroon" title="c11">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E8:54_1_1_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E10:54_1_1_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E12:54_1_1_1"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="txt" href="#E8:54_1_1_1_1"><span class="lab"><font color="Green" title="E49">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c15">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_1_1_1_1"><span class="lab"><font color="Green" title="E49">A14</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_1_1_2"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<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">X</font>,<font color="Maroon" title="c1">Y</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b4">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b6">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b5">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )</span><br/><div class="add"><a NAME="E2:54_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A16</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:54_1_1_2"/>
<font color="Maroon" title="c2">n</font> <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"><font color="Maroon" title="c2">n</font> <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="c5">n1</font> = <font color="Maroon" title="c2">n</font> <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> <font color="Maroon" title="c3">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</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="c6">Xn1</font> = <font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font> as   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a> ;<br/><a NAME="E6:54_1_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">n</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> is   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">n</font></span>)</span> implies <font color="Maroon" title="c7">x</font> is   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">n</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="c7">x</font> is   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a></span><br/>

<a NAME="E2:54_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T70" onmouseover="rs('relat_1/T70')" onmouseout="rh()">RELAT_1:70</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:54_1_1_2_1"/>
<font color="Maroon" title="c7">x</font> is   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D10" onmouseover="rs('prvect_2/D10')" onmouseout="rh()">PRVECT_2:def 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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">n</font> as   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="prvect_2.html#D10" onmouseover="rs('prvect_2/D10')" onmouseout="rh()">PRVECT_2:def 10</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">CXn1</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Xn1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">CK</font> =  <a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c7">K</font>;<br/><a NAME="E8:54_1_1_2"/>
 <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="c6">Xn1</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="c6">Xn1</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="c6">Xn1</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="c6">Xn1</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="c6">Xn1</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="c6">Xn1</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="E9:54_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A17</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="c6">Xn1</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="c6">Xn1</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="E10:54_1_1_2"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
 <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c7">K</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="c7">K</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="c7">K</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="c7">K</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="c7">K</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="c7">K</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="E11:54_1_1_2"/>
 <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">K</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="c6">Xn1</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="c7">K</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="c6">Xn1</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="c7">K</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="c6">Xn1</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="c7">K</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="c6">Xn1</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="c7">K</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="c6">Xn1</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="c7">K</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="c6">Xn1</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="E47">A19</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="c7">K</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="c6">Xn1</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="c7">K</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="c6">Xn1</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="c7">K</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="c6">Xn1</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/><span class="kw">set </span><font color="Maroon" title="c10">KY</font> =  <a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>);<br/><a NAME="E13:54_1_1_2"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
 <a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="lopban14.html#T40" target="_self" onmouseover="rs('lopban14/T40')" onmouseout="rh()">Th41</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">I0</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E15:54_1_1_2"/><span class="lab"><font color="Green" title="E49">A21</font></span>: 
( <font color="Maroon" title="c11">I0</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c11">I0</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c11">I0</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span><br/>  for <font color="Olive" title="b2">k</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="c7">K</font></span>)</span>  ex <font color="Olive" title="b3">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font> holds <br/> ex <font color="Olive" title="b5">Ki</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">Ki</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">Ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">K1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">K1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c7">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E1:54_1_1_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">I1</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E17:54_1_1_2"/><span class="lab"><font color="Green" title="E50">A22</font></span>: 
( <font color="Maroon" title="c12">I1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c12">I1</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c12">I1</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>)</span>)</span> holds  <font color="Maroon" title="c12">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I0</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">v</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:54_1_1_2"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="ref" href="lopban14.html#T36" target="_self" onmouseover="rs('lopban14/T36')" onmouseout="rh()">Th37</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">I2</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</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="c7">K</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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">Y</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E19:54_1_1_2"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
( <font color="Maroon" title="c13">I2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c13">I2</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c13">I2</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</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">I2</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">k</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="c7">K</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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">I2</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">k</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">k</font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T35" target="_self" onmouseover="rs('lopban14/T35')" onmouseout="rh()">Th36</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">I</font> = <font color="Maroon" title="c13">I2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">I1</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">I</font> = <font color="Maroon" title="c13">I2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">I1</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</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="c7">K</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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">Y</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_2.html#T1" onmouseover="rs('lopban_2/T1')" onmouseout="rh()">LOPBAN_2:1</a></span>;<br/><a NAME="E21:54_1_1_2"/><span class="lab"><font color="Green" title="E52">A24</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</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="c15">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#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">v</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="54_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c15">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</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"><font color="Maroon" title="c16">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/>

<span class="kw">thus </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">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></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="c13">I2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</font></span>)</span></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="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<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">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</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="#E19:54_1_1_2"><span class="lab"><font color="Green" title="E51">A23</font></span></a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">v</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="#E17:54_1_1_2"><span class="lab"><font color="Green" title="E50">A22</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><span class="kw">end;</span></div><a NAME="E22:54_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">I</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_7.html#T7" onmouseover="rs('ndiff_7/T7')" onmouseout="rh()">NDIFF_7:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">I</font> = <font color="Maroon" title="c15">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<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">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>;<br/><span class="kw">take </span>
<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="c16">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c16">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c16">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b3">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b5">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E24:54_1_1_2"/>
( <font color="Maroon" title="c16">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c16">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c16">I</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E13:54_1_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E17:54_1_1_2"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="txt" href="#E19:54_1_1_2"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="txt" href="#E21:54_1_1_2"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="ref" href="funct_2.html#T27" onmouseover="rs('funct_2/T27')" onmouseout="rh()">FUNCT_2:27</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">  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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b3">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b5">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E25:54_1_1_2"/>
 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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b3">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b5">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X</font></span>)</span>  ex <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b4">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b3">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b4">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b3">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b3">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c18">x</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">X</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">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b3">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b4">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b3">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b3">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b2">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c19">pk</font>, <font color="Maroon" title="c20">pxn1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:54_1_1_2_4"/><span class="lab"><font color="Green" title="E53">A25</font></span>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">pk</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c20">pxn1</font> &amp; <font color="Maroon" title="c19">pk</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="c7">K</font></span>)</span> &amp; <font color="Maroon" title="c20">pxn1</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="c6">Xn1</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="#E3:54_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E12:54_1_1_2"><span class="lab"><font color="Green" title="E47">A19</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">pk</font> = <font color="Maroon" title="c19">pk</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="c7">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54_1_1_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E2:54_1_1_2_4"><span class="lab"><font color="Green" title="E53">A25</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c6">Xn1</font><span class="kw"> such that </span><br/><a NAME="E5:54_1_1_2_4"/><span class="lab"><font color="Green" title="E54">A26</font></span>: 
<font color="Maroon" title="c20">pxn1</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">y</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="#E9:54_1_1_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E2:54_1_1_2_4"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E6:54_1_1_2_4"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c21">pk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c7">K</font></span>)</span> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">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="c7">K</font></span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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="c7">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54_1_1_2"><span class="lab"><font color="Green" title="E46">A18</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="E7:54_1_1_2_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">pk</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="c7">K</font></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/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E55">A27</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">pk</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="c7">K</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 href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</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/>.= 
<font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="finseq_3.html#T53" onmouseover="rs('finseq_3/T53')" onmouseout="rh()">FINSEQ_3:53</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">xn1</font> = <font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c6">Xn1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1_2_4"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="txt" href="#E5:54_1_1_2_4"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E8:54_1_1_2_4"><span class="lab"><font color="Green" title="E55">A27</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">reconsider </span><font color="Maroon" title="c24">u1</font> = <font color="Maroon" title="c17">u</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="c6">Xn1</font>,<span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <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="c25">uxn1</font> = <font color="Maroon" title="c24">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">xn1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">g1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E13:54_1_1_2_4"/><span class="lab"><font color="Green" title="E56">A28</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">uxn1</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font> holds <br/> ex <font color="Olive" title="b2">Ki</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Ki</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">K1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">K1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c7">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">uxn1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">pk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:54_1_1_2"><span class="lab"><font color="Green" title="E49">A21</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">g</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/>

<span class="kw">thus </span> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></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> <font color="Maroon" title="c26">g1</font></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/>.= 
1 <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> <font color="Maroon" title="c26">g1</font></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 href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E13:54_1_1_2_4"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="ref" href="finseq_3.html#T53" onmouseover="rs('finseq_3/T53')" onmouseout="rh()">FINSEQ_3:53</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"> ( <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E15:54_1_1_2_4"/><span class="lab"><font color="Green" title="E57">A29</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</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">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E16:54_1_1_2_4"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b2">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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">  ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> &amp; <font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font> implies  ex <font color="Olive" title="b1">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1_1_2_4_3"/><span class="lab"><font color="Green" title="E58">A30</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> &amp; <font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/>

<a NAME="E2:54_1_1_2_4_3"/><span class="lab"><font color="Green" title="E59">A31</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E3:54_1_1_2_4_3"/><span class="lab"><font color="Green" title="E60">A32</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_4_3_1"><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="E1:54_1_1_2_4_3_1"/>
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> or <font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_4_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_1_1_2_4_3_1_1"/><span class="lab"><font color="Green" title="E61">A33</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">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">  ex <font color="Olive" title="b1">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:54_1_1_2_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A34</font></span>: 
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E3:54_1_1_2_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A35</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><a NAME="E4:54_1_1_2_4_3_1_1"/>
<font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E5:54_1_1_2_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A36</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E3:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>, <font color="Maroon" title="c30">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c29">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:54_1_1_2_4_3_1_1"/><span class="lab"><font color="Green" title="E65">A37</font></span>: 
( <font color="Maroon" title="c29">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c30">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c26">g1</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"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2_4"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="txt" href="#E3:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E63">A35</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c29">Xi</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">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c29">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Maroon" title="c29">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c30">h</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="c29">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c30">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="lab"><font color="Green" title="E66">A39</font></span>: <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="c7">K</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
<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="c7">K</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="txt" href="#E5:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1

<br/>.= 
<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">X</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="finseq_3.html#T53" onmouseover="rs('finseq_3/T53')" onmouseout="rh()">FINSEQ_3:53</a></span>
<br/>.= 
<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">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</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="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
;<br/><a NAME="E9:54_1_1_2_4_3_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="c7">K</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E10:54_1_1_2_4_3_1_1"/><span class="kw">then </span>
<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="c7">K</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E11:54_1_1_2_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A40</font></span>: 
<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="c7">K</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="finseq_3.html#T53" onmouseover="rs('finseq_3/T53')" onmouseout="rh()">FINSEQ_3:53</a></span>;<br/><a NAME="E12:54_1_1_2_4_3_1_1"/><span class="lab"><font color="Green" title="E68">A41</font></span>: 
<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="c7">K</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">K</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><a NAME="E13:54_1_1_2_4_3_1_1"/><span class="lab"><font color="Green" title="E69">A43</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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/><a NAME="E14:54_1_1_2_4_3_1_1"/><span class="lab"><font color="Green" title="E70">A44</font></span>: 
<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#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <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/><a NAME="E15:54_1_1_2_4_3_1_1"/>
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> <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="E16:54_1_1_2_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A45</font></span>: 
<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#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></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="c28">i</font> <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="#E1:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E61">A33</font></span></a>, <a class="txt" href="#E14:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E17:54_1_1_2_4_3_1_1"/><span class="lab"><font color="Green" title="E72">A46</font></span>: 
<font color="Maroon" title="c28">i</font> <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">g1</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="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E13:54_1_1_2_4"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="lab"><font color="Green" title="E73">A47</font></span>: <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">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
<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">X</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>
;<br/><a NAME="E19:54_1_1_2_4_3_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E62">A34</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E20:54_1_1_2_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E21:54_1_1_2_4_3_1_1"/>
<font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E22:54_1_1_2_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A49</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E23:54_1_1_2_4_3_1_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> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E24:54_1_1_2_4_3_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E75">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E25:54_1_1_2_4_3_1_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E26:54_1_1_2_4_3_1_1"/><span class="kw">then </span>
<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">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</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> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="txt" href="#E20:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E74">A48</font></span></a></span>;<br/><a NAME="E27:54_1_1_2_4_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A50</font></span>: 
<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">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">pk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_1_1_2_4"><span class="lab"><font color="Green" title="E55">A27</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="lab"><font color="Green" title="E77">A51</font></span>: <font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E13:54_1_1_2_4"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E1:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E61">A33</font></span></a>, <a class="txt" href="#E14:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="finseq_1.html#T23" onmouseover="rs('finseq_1/T23')" onmouseout="rh()">FINSEQ_1:23</a></span>
;<br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c26">g1</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"><font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E16:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="txt" href="#E17:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="ref" href="finseq_1.html#T23" onmouseover="rs('finseq_1/T23')" onmouseout="rh()">FINSEQ_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c30">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1_2_4"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="txt" href="#E3:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="txt" href="#E7:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="txt" href="#E8:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E66">A39</font></span></a>, <a class="txt" href="#E27:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E76">A50</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/><span class="kw">hence </span><a NAME="E30:54_1_1_2_4_3_1_1"/>
( <font color="Maroon" title="c29">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c30">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="txt" href="#E8:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E66">A39</font></span></a>, <a class="txt" href="#E11:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E67">A40</font></span></a>, <a class="txt" href="#E12:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E68">A41</font></span></a>, <a class="txt" href="#E28:54_1_1_2_4_3_1_1"><span class="lab"><font color="Green" title="E77">A51</font></span></a>, <a class="ref" href="finseq_1.html#T82" onmouseover="rs('finseq_1/T82')" onmouseout="rh()">FINSEQ_1:82</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="54_1_1_2_4_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_1_1_2_4_3_1_2"/>
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <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">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:54_1_1_2_4_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E3:54_1_1_2_4_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A53</font></span>: 
<font color="Maroon" title="c28">i</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="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:54_1_1_2_4_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A54</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</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="E5:54_1_1_2_4_3_1_2"/><span class="lab"><font color="Green" title="E63">A55</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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/><a NAME="E6:54_1_1_2_4_3_1_2"/><span class="lab"><font color="Green" title="E64">A56</font></span>: 
<font color="Maroon" title="c28">i</font> <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">g1</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="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E13:54_1_1_2_4"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="lab"><font color="Green" title="E65">A57</font></span>: <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c26">g1</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"><font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E62">A54</font></span></a>, <a class="txt" href="#E5:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E63">A55</font></span></a>, <a class="txt" href="#E6:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E64">A56</font></span></a>, <a class="ref" href="finseq_1.html#T23" onmouseover="rs('finseq_1/T23')" onmouseout="rh()">FINSEQ_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c25">uxn1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E3:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E61">A53</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a>, <a class="txt" href="#E13:54_1_1_2_4"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>
;<br/><a NAME="E8:54_1_1_2_4_3_1_2"/><span class="lab"><font color="Green" title="E66">A58</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="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">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="c3">X</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">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">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c17">u</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="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E9:54_1_1_2_4_3_1_2"/>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</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="#E4:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E62">A54</font></span></a>, <a class="txt" href="#E8:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="ref" href="finseq_1.html#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</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="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E10:54_1_1_2_4_3_1_2"/>
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:54_1_1_2_4"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E1:54_1_1_2_4_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E3:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E61">A53</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"> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><span class="kw">thus </span><a NAME="E11:54_1_1_2_4_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E4:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E62">A54</font></span></a>, <a class="txt" href="#E7:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E65">A57</font></span></a>, <a class="txt" href="#E8:54_1_1_2_4_3_1_2"><span class="lab"><font color="Green" title="E66">A58</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">consider </span><font color="Maroon" title="c28">K1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>, <font color="Maroon" title="c29">h1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c28">K1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E18:54_1_1_2_4"/><span class="lab"><font color="Green" title="E58">A59</font></span>: 
( <font color="Maroon" title="c28">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c29">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c7">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">uxn1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">pk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2_4"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>;<br/>
<a NAME="E19:54_1_1_2_4"/><span class="lab"><font color="Green" title="E59">A60</font></span>: 
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> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1_2"><span class="lab"><font color="Green" title="E44">A16</font></span></a></span>;<br/>
<a NAME="E20:54_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A61</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></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> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></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="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E21:54_1_1_2_4"/><span class="lab"><font color="Green" title="E61">A62</font></span>: 
1 <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#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <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="c7">K</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#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</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="#E4:54_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E2:54_1_1_2"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E22:54_1_1_2_4"/><span class="lab"><font color="Green" title="E62">A63</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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="lab"><font color="Green" title="E63">A64</font></span>: <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</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="#E2:54_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="finseq_3.html#T53" onmouseover="rs('finseq_3/T53')" onmouseout="rh()">FINSEQ_3:53</a></span>
<br/>.= 
<font color="Maroon" title="c26">g1</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2_4"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="txt" href="#E21:54_1_1_2_4"><span class="lab"><font color="Green" title="E61">A62</font></span></a>, <a class="txt" href="#E22:54_1_1_2_4"><span class="lab"><font color="Green" title="E62">A63</font></span></a>, <a class="ref" href="finseq_1.html#T23" onmouseover="rs('finseq_1/T23')" onmouseout="rh()">FINSEQ_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c26">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c7">K</font></span>)</span>

;<br/>
<a NAME="E24:54_1_1_2_4"/>
1 <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="c21">pk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_1_1_2_4"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="txt" href="#E19:54_1_1_2_4"><span class="lab"><font color="Green" title="E59">A60</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="E25:54_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A65</font></span>: 
<font color="Maroon" title="c21">pk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1_2_4"><span class="lab"><font color="Green" title="E53">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="E26:54_1_1_2_4"/><span class="lab"><font color="Green" title="E65">A66</font></span>: 
<font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">u</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2"><span class="lab"><font color="Green" title="E48">A20</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="c30">v</font> = <font color="Maroon" title="c12">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">u</font> as   <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> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">sn11</font> = <font color="Maroon" title="c23">xn1</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">X</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c5">n1</font></span>)</span> ;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">I2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pk</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">xn1</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="#E2:54_1_1_2_4"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="txt" href="#E5:54_1_1_2_4"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E8:54_1_1_2_4"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="txt" href="#E26:54_1_1_2_4"><span class="lab"><font color="Green" title="E65">A66</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="p1">(<span class="default"><font color="Maroon" title="c30">v</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c31">sn11</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">pk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:54_1_1_2"><span class="lab"><font color="Green" title="E51">A23</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I0</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">u1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">xn1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">pk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E17:54_1_1_2"><span class="lab"><font color="Green" title="E50">A22</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">uxn1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">pk</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>
;<br/>
<span class="kw">hence </span><a NAME="E30:54_1_1_2_4"/>
 ex <font color="Olive" title="b1">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c17">u</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c3">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:54_1_1_2_4"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="txt" href="#E20:54_1_1_2_4"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="txt" href="#E23:54_1_1_2_4"><span class="lab"><font color="Green" title="E63">A64</font></span></a>, <a class="txt" href="#E25:54_1_1_2_4"><span class="lab"><font color="Green" title="E64">A65</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:54"/><span class="lab"><font color="Green" title="E39">A68</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E38">A2</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c2">X</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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">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="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> <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">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) &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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">X</font></span>)</span>  ex <font color="Olive" title="b4">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font> holds <br/> ex <font color="Olive" title="b6">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b5">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c2">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:54"/><span class="lab"><font color="Green" title="E40">A69</font></span>: 
( <font color="Maroon" title="c3">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c3">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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">X</font></span>)</span>  ex <font color="Olive" title="b3">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font> holds <br/> ex <font color="Olive" title="b5">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b4">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b4">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c2">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E39">A68</font></span></a>, <a class="ref" href="finseq_1.html#T20" onmouseover="rs('finseq_1/T20')" onmouseout="rh()">FINSEQ_1:20</a></span>;<br/>
<a NAME="E6:54"/>
 for <font color="Olive" title="b1">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">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="c3">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="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">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="#E5:54"><span class="lab"><font color="Green" title="E40">A69</font></span></a>, <a class="ref" href="ndiff_7.html#T7" onmouseover="rs('ndiff_7/T7')" onmouseout="rh()">NDIFF_7:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:54"/>
 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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">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="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> <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">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) &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="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">X</font></span>)</span>  ex <font color="Olive" title="b4">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font> holds <br/> ex <font color="Olive" title="b6">Xi</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b6">Xi</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b6">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) &amp;  ex <font color="Olive" title="b5">X1</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a> ex <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban14.html#K5" title="LOPBAN14:func.5">NestingLB</a> (<font color="Olive" title="b5">X1</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span> st <br/>( <font color="Olive" title="b5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c2">X</font></span>)</span> &amp; <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="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E40">A69</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/>


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