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

<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span><br/> for <font color="Olive" title="b2">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b2">x0</font> iff 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#K5" title="NUMBERS:func.5">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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b3">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b2">x0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> holds <br/> ( <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x0</font> iff 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#K5" title="NUMBERS:func.5">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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font> iff 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#K5" title="NUMBERS:func.5">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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">RB</font> =  <a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">RB1</font> =  <a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>;<br/>
<a NAME="E1:37"/>
( ( 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#K5" title="NUMBERS:func.5">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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font> ) implies <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1"/><span class="lab"><font color="Green" title="E27">AS</font></span>: 
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#K5" title="NUMBERS:func.5">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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ] <span class="kw">means </span>( $2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> &amp;  ex <font color="Olive" title="b1">Ri</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> st <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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">y</font> where <font color="Olive" title="b3">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 </span>}</span>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Ri</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> );<br/>
<a NAME="E2:37_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/> ex <font color="Olive" title="b2">dk</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">dk</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> implies  ex <font color="Olive" title="b1">dk</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>,<font color="Olive" title="b1">dk</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">dk</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>,<font color="Olive" title="b1">dk</font>]</span><br/>

<a NAME="E2:37_1_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</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="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E3:37_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E27">AS</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Nk</font> being    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x0</font><span class="kw"> such that </span><br/><a NAME="E5:37_1_1"/><span class="lab"><font color="Green" title="E29">B1</font></span>: 
( <font color="Maroon" title="c8">Nk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> &amp;  ex <font color="Olive" title="b1">Rk</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> st <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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Nk</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Rk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D7" title="NDIFF_1:def.7">NDIFF_1:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">dk</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E7:37_1_1"/><span class="lab"><font color="Green" title="E30">B2</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">dk</font> &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">dk</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Nk</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_1.html#D3" title="NFCONT_1:def.3">NFCONT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">dk</font> = <font color="Maroon" title="c9">dk</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">dk</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c10">dk</font>]</span><br/>

<span class="kw">thus </span><a NAME="E9:37_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">dk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E30">B2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">dk</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> &amp;  ex <font color="Olive" title="b1">Ri</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> st <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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">y</font> where <font color="Olive" title="b3">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">dk</font> </span>}</span>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Ri</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E10:37_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">dk</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E30">B2</font></span></a>, <a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Ri</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> st <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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">y</font> where <font color="Olive" title="b3">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">dk</font> </span>}</span>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Ri</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E11:37_1_1"/>
 ex <font color="Olive" title="b1">Rk</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> st <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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">y</font> where <font color="Olive" title="b3">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">dk</font> </span>}</span>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Rk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c11">Rk</font> being   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:37_1_1_1"/><span class="lab"><font color="Green" title="E31">B3</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Nk</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Rk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E29">B1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">Rk</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">y</font> where <font color="Olive" title="b2">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">dk</font> </span>}</span>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Rk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E3:37_1_1_1"/>
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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">y</font> where <font color="Olive" title="b2">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">dk</font> </span>}</span>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Rk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E30">B2</font></span></a>, <a class="txt" href="#E2:37_1_1_1"><span class="lab"><font color="Green" title="E31">B3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">d</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:37_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S17" title="RECDEF_1:sch.17">RECDEF_1:sch 17</a>(<a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c8">d0</font> =  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c7">d</font>;<br/>
<a NAME="E5:37_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">d</font> <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="#E4:37_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">AP1</font></span>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#D2" title="RFINSEQ2:def.2">RFINSEQ2:def 2</a></span>;<br/>
<a NAME="E7:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<a NAME="E8:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">AP2</font></span>: 
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E30">AP1</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</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#K5" title="NUMBERS:func.5">NAT</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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">y</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> ) );<br/>
<a NAME="E9:37_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">x</font></span>)</span>;<br/>
<a NAME="E1:37_1_2"/><span class="lab"><font color="Green" title="E33">P71</font></span>: 
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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/> ex <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">r</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">y</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:37_1_2"/><span class="lab"><font color="Green" title="E34">P73</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S17" title="RECDEF_1:sch.17">RECDEF_1:sch 17</a>(<a class="txt" href="#E1:37_1_2"><span class="lab"><font color="Green" title="E33">P71</font></span></a>);<br/></span>
<a NAME="E4:37_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">y</font> <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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="#E3:37_1_2"><span class="lab"><font color="Green" title="E34">P73</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">P77</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c11">y</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:37_1_2_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">y</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><br/><a NAME="E2:37_1_2_1"/><span class="kw">then </span>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c11">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>] &amp; <font color="Maroon" title="c11">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_2"><span class="lab"><font color="Green" title="E34">P73</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_1_2_1"/>
<font color="Maroon" title="c11">y</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c12">y0</font> = <font color="Maroon" title="c11">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E8:37_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font> st <br/>( <font color="Maroon" title="c12">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</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#K5" title="NUMBERS:func.5">NAT</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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">y</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">x</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_2"><span class="lab"><font color="Green" title="E35">P77</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:37_1_2"/>
 ex <font color="Olive" title="b1">y0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y0</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">L</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:37_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E9:37_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>);<br/></span>
<a NAME="E12:37_1"/><span class="lab"><font color="Green" title="E34">AD1</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> holds  <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">Lx</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font><span class="kw"> such that </span><br/><a NAME="E2:37_1_3"/><span class="lab"><font color="Green" title="E35">B1</font></span>: 
( <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Lx</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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c12">Lx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">x</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">Ly</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font><span class="kw"> such that </span><br/><a NAME="E4:37_1_3"/><span class="lab"><font color="Green" title="E36">B2</font></span>: 
( <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Ly</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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c13">Ly</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c11">y</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">Lxy</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font><span class="kw"> such that </span><br/><a NAME="E6:37_1_3"/><span class="lab"><font color="Green" title="E37">B3</font></span>: 
( <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Lxy</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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<a NAME="E7:37_1_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Lxy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="euclid.html#T3" title="EUCLID:th.3">EUCLID:3</a></span>;<br/>
<a NAME="E8:37_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">B8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Lxy</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"><font color="Maroon" title="c12">Lx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c13">Ly</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T3" title="EUCLID:th.3">EUCLID:3</a></span>;<br/>
<a NAME="E9:37_1_3"/>
for <font color="Olive" title="b1">i0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Lxy</font> holds <br/><font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Lx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c13">Ly</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">i0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Lxy</font> implies <font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Lx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c13">Ly</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c15">i0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:37_1_3_1"/>
<font color="Maroon" title="c15">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Lxy</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Lx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c13">Ly</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font></span><br/>

<a NAME="E3:37_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T3" title="EUCLID:th.3">EUCLID:3</a></span>;<br/>
<a NAME="E4:37_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">B7</font></span>: 
( <font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">Lx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">x</font></span>)</span> &amp; <font color="Maroon" title="c13">Ly</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c11">y</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_3"><span class="lab"><font color="Green" title="E35">B1</font></span></a>, <a class="txt" href="#E4:37_1_3"><span class="lab"><font color="Green" title="E36">B2</font></span></a>, <a class="txt" href="#E6:37_1_3"><span class="lab"><font color="Green" title="E37">B3</font></span></a></span>;<br/>
<a NAME="E5:37_1_3_1"/>
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></span>;<br/>
<a NAME="E6:37_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">B6</font></span>: 
<span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c11">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">Diffxy</font> = <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">Diffx</font> = <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">Diffy</font> = <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c11">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E10:37_1_3_1"/>
<font color="Maroon" title="c17">Diffxy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Diffx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c19">Diffy</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_3_1"><span class="lab"><font color="Green" title="E40">B6</font></span></a>, <a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</a></span>;<br/>
<a NAME="E11:37_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Diffx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c19">Diffy</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_3_1"><span class="lab"><font color="Green" title="E39">B7</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>;<br/>
<a NAME="E12:37_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Diffx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Diffy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T27" title="RVSUM_1:th.27">RVSUM_1:27</a></span>;<br/>
<a NAME="E13:37_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Lx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Diffy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_3_1"><span class="lab"><font color="Green" title="E39">B7</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>;<br/>
<a NAME="E14:37_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Lx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Ly</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_3_1"><span class="lab"><font color="Green" title="E39">B7</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:37_1_3_1"/>
<font color="Maroon" title="c14">Lxy</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Lx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c13">Ly</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T27" title="RVSUM_1:th.27">RVSUM_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:37_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">Lxy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Lx</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c13">Ly</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_3"><span class="lab"><font color="Green" title="E38">B8</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:37_1_3"/>
<font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_3"><span class="lab"><font color="Green" title="E35">B1</font></span></a>, <a class="txt" href="#E4:37_1_3"><span class="lab"><font color="Green" title="E36">B2</font></span></a>, <a class="txt" href="#E6:37_1_3"><span class="lab"><font color="Green" title="E37">B3</font></span></a>, <a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:37_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span><br/> for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c12">Lx</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font><span class="kw"> such that </span><br/><a NAME="E2:37_1_4"/><span class="lab"><font color="Green" title="E35">B1</font></span>: 
( <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Lx</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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c12">Lx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">x</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">Lrx</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font><span class="kw"> such that </span><br/><a NAME="E4:37_1_4"/><span class="lab"><font color="Green" title="E36">B2</font></span>: 
( <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Lrx</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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<a NAME="E5:37_1_4"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="euclid.html#T3" title="EUCLID:th.3">EUCLID:3</a></span>;<br/>
<a NAME="E6:37_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">B3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">Lrx</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T3" title="EUCLID:th.3">EUCLID:3</a></span>;<br/>
<a NAME="E7:37_1_4"/>
for <font color="Olive" title="b1">i0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">i</font> = <font color="Maroon" title="c14">i0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:37_1_4_1"/>
<font color="Maroon" title="c14">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font></span><br/>

<a NAME="E3:37_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T3" title="EUCLID:th.3">EUCLID:3</a></span>;<br/>
<a NAME="E4:37_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">B7</font></span>: 
( <font color="Maroon" title="c12">Lx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">x</font></span>)</span> &amp; <font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_4"><span class="lab"><font color="Green" title="E35">B1</font></span></a>, <a class="txt" href="#E4:37_1_4"><span class="lab"><font color="Green" title="E36">B2</font></span></a></span>;<br/>
<a NAME="E5:37_1_4_1"/>
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></span>;<br/>
<a NAME="E6:37_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">B5</font></span>: 
<span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D6" title="LOPBAN_1:def.6">LOPBAN_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">Diffrx</font> = <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">Diffx</font> = <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E9:37_1_4_1"/>
<font color="Maroon" title="c16">Diffrx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c17">Diffx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_4_1"><span class="lab"><font color="Green" title="E39">B5</font></span></a>, <a class="ref" href="real_ns1.html#T3" title="REAL_NS1:th.3">REAL_NS1:3</a></span>;<br/>
<a NAME="E10:37_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c17">Diffx</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_4_1"><span class="lab"><font color="Green" title="E38">B7</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>;<br/>
<a NAME="E11:37_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Diffx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T67" title="RVSUM_1:th.67">RVSUM_1:67</a></span>;<br/>
<a NAME="E12:37_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Lx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_4_1"><span class="lab"><font color="Green" title="E38">B7</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:37_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Lrx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T67" title="RVSUM_1:th.67">RVSUM_1:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:37_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c12">Lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Lrx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_4"><span class="lab"><font color="Green" title="E37">B3</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:37_1_4"/>
<font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_4"><span class="lab"><font color="Green" title="E35">B1</font></span></a>, <a class="txt" href="#E4:37_1_4"><span class="lab"><font color="Green" title="E36">B2</font></span></a>, <a class="ref" href="real_ns1.html#T3" title="REAL_NS1:th.3">REAL_NS1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c10">L</font> = <font color="Maroon" title="c9">L</font> as   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1"><span class="lab"><font color="Green" title="E34">AD1</font></span></a>, <a class="ref" href="lopban_1.html#D6" title="LOPBAN_1:def.6">LOPBAN_1:def 6</a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">L0</font> = <font color="Maroon" title="c10">L</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"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">Rk</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Rk</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">y</font> where <font color="Olive" title="b3">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 </span>}</span>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Rk</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> ) );<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">$1 <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">R</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E17:37_1"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> holds  <font color="Maroon" title="c12">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E18:37_1"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span><br/> 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#K5" title="NUMBERS:func.5">NAT</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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_5"/><span class="lab"><font color="Green" title="E37">BS1</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_5"/><span class="lab"><font color="Green" title="E38">BS2</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><br/>

<a NAME="E3:37_1_5"/>
<font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>;<br/>
<a NAME="E4:37_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E5:37_1_5"/><span class="kw">then </span>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T1" title="REAL_NS1:th.1">REAL_NS1:1</a></span>;<br/>
<a NAME="E6:37_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">P1</font></span>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T10" title="EUCLID:th.10">EUCLID:10</a></span>;<br/>
<a NAME="E7:37_1_5"/><span class="lab"><font color="Green" title="E40">P2</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_5"><span class="lab"><font color="Green" title="E37">BS1</font></span></a>, <a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<a NAME="E8:37_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_5"><span class="lab"><font color="Green" title="E39">P1</font></span></a></span>;<br/>
<a NAME="E9:37_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">P3</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_5"><span class="lab"><font color="Green" title="E40">P2</font></span></a></span>;<br/>
<a NAME="E10:37_1_5"/><span class="lab"><font color="Green" title="E42">P4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">gxx0</font> = <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span>, <font color="Maroon" title="c16">gx0</font> = <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font>, <font color="Maroon" title="c17">Lx</font> = <font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">gxx01</font> = <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span>, <font color="Maroon" title="c19">gx01</font> = <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font>, <font color="Maroon" title="c20">Lx1</font> = <font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> ;<br/>
<a NAME="E13:37_1_5"/><span class="lab"><font color="Green" title="E43">B0</font></span>: 
(  <a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c16">gx0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> &amp;  <a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c17">Lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T3" title="REAL_NS1:th.3">REAL_NS1:3</a></span>;<br/>
<a NAME="E14:37_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c15">gxx0</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c16">gx0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</a></span>;<br/>
<a NAME="E15:37_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">B2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">gxx0</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c16">gx0</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c17">Lx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37_1_5"><span class="lab"><font color="Green" title="E43">B0</font></span></a>, <a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</a></span>;<br/>
<a NAME="E16:37_1_5"/>
<font color="Maroon" title="c15">gxx0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c16">gx0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>;<br/>
<a NAME="E17:37_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">B1</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">gxx0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c16">gx0</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">Lx</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>;<br/>
<a NAME="E18:37_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E19:37_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:37_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a></span>;<br/>
<a NAME="E20:37_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:37_1_5"><span class="lab"><font color="Green" title="E45">B1</font></span></a>, <a class="txt" href="#E15:37_1_5"><span class="lab"><font color="Green" title="E44">B2</font></span></a>, <a class="ref" href="pdiff_6.html#T26" target="_self" title="PDIFF_6:th.26">DPREP1201</a></span>;<br/>
<a NAME="E21:37_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">B3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_6.html#T26" target="_self" title="PDIFF_6:th.26">DPREP1201</a></span>;<br/>
<a NAME="E22:37_1_5"/>
( <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c20">Lx1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">Lx1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_6.html#T27" target="_self" title="PDIFF_6:th.27">DPREP1202</a></span>;<br/>
<a NAME="E23:37_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:37_1_5"><span class="lab"><font color="Green" title="E46">B3</font></span></a>, <a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>;<br/>
<a NAME="E24:37_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>;<br/>
<a NAME="E25:37_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></span>;<br/>
<a NAME="E26:37_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">C0</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></span>;<br/>
<a NAME="E27:37_1_5"/>
( <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> )
 
;<br/>
<a NAME="E28:37_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">C1</font></span>: 
( <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E49">C2</font></span>: <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_5"><span class="lab"><font color="Green" title="E38">BS2</font></span></a>, <a class="txt" href="#E28:37_1_5"><span class="lab"><font color="Green" title="E48">C1</font></span></a>, <a class="ref" href="partfun2.html#T9" title="PARTFUN2:th.9">PARTFUN2:9</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_5"><span class="lab"><font color="Green" title="E41">P3</font></span></a>, <a class="txt" href="#E10:37_1_5"><span class="lab"><font color="Green" title="E42">P4</font></span></a>, <a class="txt" href="#E28:37_1_5"><span class="lab"><font color="Green" title="E48">C1</font></span></a>, <a class="ref" href="partfun2.html#T9" title="PARTFUN2:th.9">PARTFUN2:9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E31:37_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:37_1_5"><span class="lab"><font color="Green" title="E47">C0</font></span></a>, <a class="txt" href="#E29:37_1_5"><span class="lab"><font color="Green" title="E49">C2</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:37_1"/>
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b3">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b3">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b3">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">d</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b3">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b3">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">d</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b2">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_6"/><span class="lab"><font color="Green" title="E37">B1</font></span>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">d</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b2">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c14">r0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font>;<br/>
<a NAME="E2:37_1_6"/><span class="lab"><font color="Green" title="E38">DD0</font></span>: 
 <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T93" title="SQUARE_1:th.93">SQUARE_1:93</a></span>;<br/>
<a NAME="E3:37_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">DE0</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_6"><span class="lab"><font color="Green" title="E37">B1</font></span></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ] <span class="kw">means </span>( $2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> $1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font> ) );<br/>
<a NAME="E4:37_1_6"/><span class="lab"><font color="Green" title="E40">DD1</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/> ex <font color="Olive" title="b2">dd</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">dd</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> implies  ex <font color="Olive" title="b1">dd</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">k</font>,<font color="Olive" title="b1">dd</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_6_1"/><span class="lab"><font color="Green" title="E41">D0</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">dd</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">k</font>,<font color="Olive" title="b1">dd</font>]</span><br/>

<a NAME="E2:37_1_6_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</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="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E3:37_1_6_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E27">AS</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">Nk</font> being    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x0</font><span class="kw"> such that </span><br/><a NAME="E5:37_1_6_1"/><span class="lab"><font color="Green" title="E42">D1</font></span>: 
( <font color="Maroon" title="c16">Nk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> &amp;  ex <font color="Olive" title="b1">PR</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> st <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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Nk</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D7" title="NDIFF_1:def.7">NDIFF_1:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">d0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E7:37_1_6_1"/><span class="lab"><font color="Green" title="E43">D2</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">d0</font> &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">d0</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Nk</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_1.html#D3" title="NFCONT_1:def.3">NFCONT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">PR</font> being   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span><span class="kw"> such that </span><br/><a NAME="E9:37_1_6_1"/><span class="lab"><font color="Green" title="E44">D3</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Nk</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">PR</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_6_1"><span class="lab"><font color="Green" title="E42">D1</font></span></a></span>;<br/>
<a NAME="E10:37_1_6_1"/>
<font color="Maroon" title="c18">PR</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" title="NDIFF_1:def.5">NDIFF_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">d1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E12:37_1_6_1"/><span class="lab"><font color="Green" title="E45">D4</font></span>: 
( <font color="Maroon" title="c19">d1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">d1</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">PR</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_6"><span class="lab"><font color="Green" title="E39">DE0</font></span></a>, <a class="ref" href="ndiff_1.html#T26" title="NDIFF_1:th.26">NDIFF_1:26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">dd</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font>;<br/>
<span class="kw">take </span>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">k</font>, <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font>]</span><br/>

<a NAME="E13:37_1_6_1"/>
for <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_6_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font> implies <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c21">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_6_1_1"/><span class="lab"><font color="Green" title="E46">D6</font></span>: 
( <font color="Maroon" title="c21">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c21">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span><br/>

<a NAME="E2:37_1_6_1_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<a NAME="E3:37_1_6_1_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_6_1_1"><span class="lab"><font color="Green" title="E46">D6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:37_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">D7</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">PR</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_6_1_1"><span class="lab"><font color="Green" title="E46">D6</font></span></a>, <a class="txt" href="#E12:37_1_6_1"><span class="lab"><font color="Green" title="E45">D4</font></span></a></span>;<br/>
<a NAME="E5:37_1_6_1_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">d0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<a NAME="E6:37_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">D8</font></span>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">d0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_6_1_1"><span class="lab"><font color="Green" title="E46">D6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:37_1_6_1_1"/><span class="lab"><font color="Green" title="E49">D9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_4.html#T1" title="RLVECT_4:th.1">RLVECT_4:1</a></span>;<br/>
<a NAME="E8:37_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">E0</font></span>: 
<font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">d0</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_6_1_1"><span class="lab"><font color="Green" title="E48">D8</font></span></a></span>;<br/>
<a NAME="E9:37_1_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Nk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_6_1"><span class="lab"><font color="Green" title="E43">D2</font></span></a></span>;<br/>
<a NAME="E10:37_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">E2</font></span>: 
<font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_6_1"><span class="lab"><font color="Green" title="E42">D1</font></span></a></span>;<br/>
<a NAME="E11:37_1_6_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">PR</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_6_1_1"><span class="lab"><font color="Green" title="E50">E0</font></span></a>, <a class="txt" href="#E7:37_1_6_1"><span class="lab"><font color="Green" title="E43">D2</font></span></a>, <a class="txt" href="#E9:37_1_6_1"><span class="lab"><font color="Green" title="E44">D3</font></span></a></span>;<br/>
<a NAME="E12:37_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">D10</font></span>: 
<font color="Maroon" title="c18">PR</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c21">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_6_1_1"><span class="lab"><font color="Green" title="E49">D9</font></span></a>, <a class="ref" href="rlvect_4.html#T1" title="RLVECT_4:th.1">RLVECT_4:1</a></span>;<br/>
<a NAME="E13:37_1_6_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></span>;<br/>
<a NAME="E14:37_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">E3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1_6_1_1"><span class="lab"><font color="Green" title="E51">E2</font></span></a>, <a class="txt" href="#E1:37_1_6_1"><span class="lab"><font color="Green" title="E41">D0</font></span></a>, <a class="txt" href="#E18:37_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font><span class="kw"> such that </span><br/><a NAME="E16:37_1_6_1_1"/><span class="lab"><font color="Green" title="E54">D11</font></span>: 
( <font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c22">y</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c21">z</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<a NAME="E17:37_1_6_1_1"/><span class="lab"><font color="Green" title="E55">D12</font></span>: 
<font color="Maroon" title="c22">y</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c21">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_1_6_1_1"><span class="lab"><font color="Green" title="E54">D11</font></span></a>, <a class="txt" href="#E1:37_1_6_1"><span class="lab"><font color="Green" title="E41">D0</font></span></a></span>;<br/>
<a NAME="E18:37_1_6_1_1"/>
<span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c21">z</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">Dk</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E20:37_1_6_1_1"/><span class="lab"><font color="Green" title="E56">D13</font></span>: 
<span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c21">z</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="c23">Dk</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="ref" href="finseq_2.html#T117" title="FINSEQ_2:th.117">FINSEQ_2:117</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">y1</font> = <font color="Maroon" title="c22">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E22:37_1_6_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E23:37_1_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E24:37_1_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">y1</font></span>)</span></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="#E16:37_1_6_1_1"><span class="lab"><font color="Green" title="E54">D11</font></span></a>, <a class="ref" href="pdiff_1.html#D4" title="PDIFF_1:def.4">PDIFF_1:def 4</a></span>;<br/>
<a NAME="E25:37_1_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">z</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c21">z</font></span>)</span></span>)</span></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="#E17:37_1_6_1_1"><span class="lab"><font color="Green" title="E55">D12</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:37_1_6_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c21">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_6_1_1"><span class="lab"><font color="Green" title="E47">D7</font></span></a>, <a class="txt" href="#E12:37_1_6_1_1"><span class="lab"><font color="Green" title="E52">D10</font></span></a>, <a class="txt" href="#E14:37_1_6_1_1"><span class="lab"><font color="Green" title="E53">E3</font></span></a>, <a class="txt" href="#E20:37_1_6_1_1"><span class="lab"><font color="Green" title="E56">D13</font></span></a>, <a class="ref" href="pdiff_1.html#T1" title="PDIFF_1:th.1">PDIFF_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:37_1_6_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">k</font>, <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c17">d0</font>,<font color="Maroon" title="c19">d1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_6_1"><span class="lab"><font color="Green" title="E43">D2</font></span></a>, <a class="txt" href="#E12:37_1_6_1"><span class="lab"><font color="Green" title="E45">D4</font></span></a>, <a class="ref" href="xxreal_0.html#T21" title="XXREAL_0:th.21">XXREAL_0:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c15">dd</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E6:37_1_6"/><span class="lab"><font color="Green" title="E41">DD2</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">dd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c15">dd</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S17" title="RECDEF_1:sch.17">RECDEF_1:sch 17</a>(<a class="txt" href="#E4:37_1_6"><span class="lab"><font color="Green" title="E40">DD1</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c16">d</font> =  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font>;<br/>
<span class="kw">take </span>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> ) )</span><br/>

<a NAME="E7:37_1_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">dd</font> <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="#E6:37_1_6"><span class="lab"><font color="Green" title="E41">DD2</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:37_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">PP1</font></span>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c15">dd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">dd</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#D2" title="RFINSEQ2:def.2">RFINSEQ2:def 2</a></span>;<br/>
<a NAME="E9:37_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">DD3</font></span>: 
<font color="Maroon" title="c15">dd</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c15">dd</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_6"><span class="lab"><font color="Green" title="E41">DD2</font></span></a></span>;<br/>
<a NAME="E10:37_1_6"/>
for <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> implies <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c17">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_6_2"/><span class="lab"><font color="Green" title="E44">G0</font></span>: 
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c17">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">Rz</font> = <font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c17">z</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">R0</font> = <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T132" title="FINSEQ_2:th.132">FINSEQ_2:132</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">SRz</font> =  <a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E5:37_1_6_2"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_6_2"><span class="lab"><font color="Green" title="E44">G0</font></span></a>, <a class="ref" href="normsp_1.html#D2" title="NORMSP_1:def.2">NORMSP_1:def 2</a></span>;<br/>
<a NAME="E6:37_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">GG4</font></span>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a></span>;<br/>
<a NAME="E7:37_1_6_2"/><span class="lab"><font color="Green" title="E46">DD4</font></span>: 
for <font color="Olive" title="b1">i0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_6_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i0</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c21">i0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:37_1_6_2_1"/><span class="lab"><font color="Green" title="E47">G1</font></span>: 
<font color="Maroon" title="c21">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i0</font></span><br/>

<a NAME="E3:37_1_6_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">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> <font color="Maroon" title="c18">Rz</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/>
<a NAME="E4:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E5:37_1_6_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">G3</font></span>: 
<span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D2" title="RVSUM_1:def.2">RVSUM_1:def 2</a></span>;<br/>
<a NAME="E6:37_1_6_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">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="#E2:37_1_6_2_1"><span class="lab"><font color="Green" title="E47">G1</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:37_1_6_2_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">dd</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_6"><span class="lab"><font color="Green" title="E41">DD2</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:37_1_6_2_1"/><span class="kw">then </span>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">dd</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#T2" title="RFINSEQ2:th.2">RFINSEQ2:2</a></span>;<br/>
<a NAME="E9:37_1_6_2_1"/><span class="kw">then </span>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">dd</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_6_2_1"><span class="lab"><font color="Green" title="E47">G1</font></span></a>, <a class="txt" href="#E6:37_1_6"><span class="lab"><font color="Green" title="E41">DD2</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E10:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">dd</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_6_2"><span class="lab"><font color="Green" title="E44">G0</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c17">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_6_2"><span class="lab"><font color="Green" title="E44">G0</font></span></a>, <a class="txt" href="#E2:37_1_6_2_1"><span class="lab"><font color="Green" title="E47">G1</font></span></a>, <a class="txt" href="#E6:37_1_6"><span class="lab"><font color="Green" title="E41">DD2</font></span></a></span>;<br/>
<a NAME="E12:37_1_6_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">G5</font></span>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c17">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_6_2"><span class="lab"><font color="Green" title="E45">GG4</font></span></a>, <a class="ref" href="xreal_1.html#T83" title="XREAL_1:th.83">XREAL_1:83</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">Rzi</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T118" title="FINSEQ_2:th.118">FINSEQ_2:118</a></span>;<br/>
<a NAME="E14:37_1_6_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E15:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">z</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">Rz</font></span>)</span></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="ref" href="pdiff_1.html#D4" title="PDIFF_1:def.4">PDIFF_1:def 4</a></span>;<br/>
<a NAME="E16:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">z</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"><span class="p2">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span></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="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>;<br/>
<a NAME="E17:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c23">Rzi</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T1" title="REAL_NS1:th.1">REAL_NS1:1</a></span>;<br/>
<a NAME="E18:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1_6_2_1"><span class="lab"><font color="Green" title="E49">G5</font></span></a>, <a class="ref" href="jordan2c.html#T12" title="JORDAN2C:th.12">JORDAN2C:12</a></span>;<br/>
<a NAME="E19:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a>, <a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E20:37_1_6_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:37_1_6_2_1"/>
<font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_6_2_1"><span class="lab"><font color="Green" title="E47">G1</font></span></a>, <a class="txt" href="#E5:37_1_6_2_1"><span class="lab"><font color="Green" title="E48">G3</font></span></a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:37_1_6_2"/><span class="lab"><font color="Green" title="E47">DD8</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_6_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_6_2_2"/>
<font color="Maroon" title="c21">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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span><br/>

<a NAME="E2:37_1_6_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Rz</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">Rz</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></span>;<br/>
<a NAME="E3:37_1_6_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E4:37_1_6_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Rz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D2" title="RVSUM_1:def.2">RVSUM_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:37_1_6_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:37_1_6_2"/><span class="lab"><font color="Green" title="E48">DD9</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/>
<a NAME="E10:37_1_6_2"/><span class="lab"><font color="Green" title="E49">DD5</font></span>: 
 ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_6_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 )</span><br/>

<a NAME="E1:37_1_6_2_3"/>
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" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:37_1_6_2_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span><br/>

<span class="kw">hence </span><a NAME="E3:37_1_6_2_3"/>
<font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_6_2"><span class="lab"><font color="Green" title="E46">DD4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:37_1_6_2"/><span class="lab"><font color="Green" title="E50">LP1</font></span>: 
 <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T93" title="SQUARE_1:th.93">SQUARE_1:93</a></span>;<br/>
<a NAME="E12:37_1_6_2"/>
for <font color="Olive" title="b1">i0</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c20">SRz</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">R0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_6_2"><span class="lab"><font color="Green" title="E46">DD4</font></span></a></span>;<br/>
<a NAME="E13:37_1_6_2"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c20">SRz</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">R0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1_6_2"><span class="lab"><font color="Green" title="E49">DD5</font></span></a>, <a class="ref" href="rvsum_1.html#T113" title="RVSUM_1:th.113">RVSUM_1:113</a></span>;<br/>
<a NAME="E14:37_1_6_2"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c18">Rz</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></span>;<br/>
<a NAME="E15:37_1_6_2"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">Rz</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_6_2"><span class="lab"><font color="Green" title="E47">DD8</font></span></a>, <a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a>, <a class="ref" href="rvsum_1.html#T114" title="RVSUM_1:th.114">RVSUM_1:114</a></span>;<br/>
<a NAME="E16:37_1_6_2"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">Rz</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_6_2"><span class="lab"><font color="Green" title="E48">DD9</font></span></a>, <a class="ref" href="square_1.html#T97" title="SQUARE_1:th.97">SQUARE_1:97</a></span>;<br/>
<a NAME="E17:37_1_6_2"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">Rz</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T158" title="COMPLEX1:th.158">COMPLEX1:158</a></span>;<br/>
<a NAME="E18:37_1_6_2"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">Rz</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_6"><span class="lab"><font color="Green" title="E38">DD0</font></span></a>, <a class="txt" href="#E1:37_1_6"><span class="lab"><font color="Green" title="E37">B1</font></span></a>, <a class="txt" href="#E6:37_1_6_2"><span class="lab"><font color="Green" title="E45">GG4</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E19:37_1_6_2"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">Rz</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
;<br/>
<a NAME="E20:37_1_6_2"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">Rz</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_6_2"><span class="lab"><font color="Green" title="E45">GG4</font></span></a>, <a class="ref" href="xreal_1.html#T85" title="XREAL_1:th.85">XREAL_1:85</a></span>;<br/>
<a NAME="E21:37_1_6_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c17">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T1" title="REAL_NS1:th.1">REAL_NS1:1</a></span>;<br/>
<a NAME="E22:37_1_6_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c17">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1_6_2"><span class="lab"><font color="Green" title="E50">LP1</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:37_1_6_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c17">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c17">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:37_1_6"/>
(  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c15">dd</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_6"><span class="lab"><font color="Green" title="E43">DD3</font></span></a>, <a class="txt" href="#E8:37_1_6"><span class="lab"><font color="Green" title="E42">PP1</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">R</font> = <font color="Maroon" title="c12">R</font> as   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#T26" title="NDIFF_1:th.26">NDIFF_1:26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">N</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c7">d</font> </span>}</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">N</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c7">d</font> </span>}</span>  as    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1"><span class="lab"><font color="Green" title="E31">AP2</font></span></a>, <a class="ref" href="nfcont_1.html#T3" title="NFCONT_1:th.3">NFCONT_1:3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font> implies <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> )</span><br/><span class="kw">assume </span><a NAME="E1:37_1_7"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:37_1_7"/><span class="lab"><font color="Green" title="E37">T2</font></span>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c7">d</font> )
 ;<br/><a NAME="E4:37_1_7"/>
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" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E5:37_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">T5</font></span>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/><a NAME="E6:37_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">T3</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> 1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/><a NAME="E7:37_1_7"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> 1,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></span>;<br/><a NAME="E8:37_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">T4</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_7"><span class="lab"><font color="Green" title="E39">T3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E9:37_1_7"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">d</font> <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="#E4:37_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E10:37_1_7"/><span class="kw">then </span>
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="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E11:37_1_7"/><span class="kw">then </span>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#T2" title="RFINSEQ2:th.2">RFINSEQ2:2</a></span>;<br/><a NAME="E12:37_1_7"/><span class="kw">then </span>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_7"><span class="lab"><font color="Green" title="E38">T5</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E13:37_1_7"/><span class="kw">then </span>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_7"><span class="lab"><font color="Green" title="E37">T2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:37_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 </span>}</span> 
 ;<br/><span class="kw">hence </span><a NAME="E15:37_1_7"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_7"><span class="lab"><font color="Green" title="E37">T2</font></span></a>, <a class="txt" href="#E8:37_1_7"><span class="lab"><font color="Green" title="E40">T4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E23:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">T1</font></span>: 
<font color="Maroon" title="c15">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E24:37_1"/>
 ex <font color="Olive" title="b1">L</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"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> ex <font color="Olive" title="b2">R</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> st <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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c11">L0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">R</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span> st <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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">R</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c16">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:37_1_8_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><br/><a NAME="E2:37_1_8_1"/><span class="lab"><font color="Green" title="E38">PP1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><a NAME="E3:37_1_8_1"/>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:37_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a></span>;<br/><a NAME="E4:37_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></span>;<br/><a NAME="E5:37_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></span>;<br/><a NAME="E6:37_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p5">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/><a NAME="E7:37_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></span>;<br/><a NAME="E8:37_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></span>;<br/><a NAME="E9:37_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_8_1"><span class="lab"><font color="Green" title="E38">PP1</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E10:37_1_8_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/><a NAME="E11:37_1_8_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/><span class="kw">hence </span><a NAME="E12:37_1_8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">L0</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E25:37_1"/>
<font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37_1"><span class="lab"><font color="Green" title="E37">T1</font></span></a>, <a class="ref" href="ndiff_1.html#D6" title="NDIFF_1:def.6">NDIFF_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:37"/>
( <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font> iff 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#K5" title="NUMBERS:func.5">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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_6.html#T28" target="_self" title="PDIFF_6:th.28">DPREP120</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
