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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">q</font> be   <a href="hahnban.html#NM2" title="HAHNBAN:NM.2">Banach-Functional</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">fi</font> be   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c2">X</font>;<br/>







<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E42">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c4">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ] <b>means </b> ex <font color="Olive" title="b1">Y</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">Y</font> &amp; $1 <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font> &amp;  ex <font color="Olive" title="b2">f'</font> being  <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Olive" title="b1">Y</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f'</font> &amp; ( for <font color="Olive" title="b3">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Olive" title="b1">Y</font><br/> for <font color="Olive" title="b4">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> holds <br/><font color="Olive" title="b2">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">v</font> ) ) );<br/>
<b>reconsider </b><font color="Maroon" title="c5">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>] </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E3:42"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/>
<a NAME="E4:42"/><b>then </b>
( <font color="Maroon" title="c4">fi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp;  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a>, <a class="ref" href="partfun1.html#T125" title="PARTFUN1:th.125">PARTFUN1:125</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">fi'</font> = <font color="Maroon" title="c4">fi</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  ;<br/>
<a NAME="E6:42"/>
<font color="Maroon" title="c6">fi'</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T40" title="FUNCT_2:th.40">FUNCT_2:40</a></i>;<br/>
<a NAME="E7:42"/><b>then </b>
<font color="Maroon" title="c4">fi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E1:42"><i><font color="Green" title="E42">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c5">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span> ;<br/>
<b>defpred </b><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> ] <b>means </b>$1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $2;<br/>
<a NAME="E9:42"/><i><font color="Green" title="E43">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>]
 
;<br/>
<a NAME="E10:42"/><i><font color="Green" title="E44">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:42"/><i><font color="Green" title="E45">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green" title="E46">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:42_1"/><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font>
 <b>and </b><br/><a NAME="E2:42_1"/><i><font color="Green" title="E48">A7</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>]
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1"/>
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1"/><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c9">u</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font>;<br/><b>take </b><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c9">u</font>;<br/><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font>;<br/><b>assume </b><a NAME="E3:42_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/><b>hence </b><a NAME="E4:42_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">u</font>]
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1_1_1"><i><font color="Green" title="E49">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_2"/><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_1_2"/><i><font color="Green" title="E50">A10</font></i>: 
<font color="Maroon" title="c8">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">E</font> = <font color="Maroon" title="c8">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="hahnban.html#T4" target="_self" title="HAHNBAN:th.4">Th4</a></i>;<br/><a NAME="E4:42_1_1_2"/>
<font color="Maroon" title="c8">B</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ;<br/>

<b>hence </b><a NAME="E2:42_1_1_2_1"/>
( <font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">y</font> or <font color="Maroon" title="c11">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="hahnban.html#E2:42_1"><i><font color="Green" title="E48">A7</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">u</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="txt" href="hahnban.html#E2:42_1_1_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="hahnban.html#T13" target="_self" title="HAHNBAN:th.13">Th13</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">B</font>;<br/><div><i><font color="Green" title="E51">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1_2_2"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_2"/><i><font color="Green" title="E52">A12</font></i>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_2"/><i><font color="Green" title="E53">A13</font></i>: 
 ex <font color="Olive" title="b1">Y</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c13">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font> &amp;  ex <font color="Olive" title="b2">f'</font> being  <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Olive" title="b1">Y</font> st <br/>( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f'</font> &amp; ( for <font color="Olive" title="b3">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Olive" title="b1">Y</font><br/> for <font color="Olive" title="b4">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> holds <br/><font color="Olive" title="b2">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">v</font> ) ) )
 ;<br/><b>consider </b><font color="Maroon" title="c14">Y</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:42_1_1_2_2"/><i><font color="Green" title="E54">A14</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">Y</font>
 <b>and </b><br/><a NAME="E7:42_1_1_2_2"/><i><font color="Green" title="E55">A15</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_2"/><i><font color="Green" title="E56">A16</font></i>: 
 ex <font color="Olive" title="b1">f'</font> being  <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c14">Y</font> st <br/>( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f'</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c14">Y</font><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> holds <br/><font color="Olive" title="b1">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</font> ) )
 <b>by </b><i><a class="txt" href="hahnban.html#E4:42_1_1_2_2"><i><font color="Green" title="E53">A13</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">f'</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c14">Y</font><b> such that </b><br/><a NAME="E10:42_1_1_2_2"/><i><font color="Green" title="E57">A17</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f'</font>
 <b>and </b><br/><a NAME="E11:42_1_1_2_2"/><i><font color="Green" title="E58">A18</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c14">Y</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c15">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E8:42_1_1_2_2"><i><font color="Green" title="E56">A16</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c16">Y</font> = <font color="Maroon" title="c14">Y</font>;<br/><b>reconsider </b><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c15">f'</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c16">Y</font> ;<br/><b>take </b><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c17">f</font>;<br/><b>thus </b><a NAME="E13:42_1_1_2_2"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E3:42_1_1_2_2"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="hahnban.html#E10:42_1_1_2_2"><i><font color="Green" title="E57">A17</font></i></a></i>;<br/><b>thus </b><a NAME="E14:42_1_1_2_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">Y</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E6:42_1_1_2_2"><i><font color="Green" title="E54">A14</font></i></a></i>;<br/><b>thus </b><a NAME="E15:42_1_1_2_2"/>
<font color="Maroon" title="c18">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2_2"><i><font color="Green" title="E55">A15</font></i></a>, <a class="txt" href="hahnban.html#E10:42_1_1_2_2"><i><font color="Green" title="E57">A17</font></i></a></i>;<br/><b>thus </b><a NAME="E16:42_1_1_2_2"/>
for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c16">Y</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E11:42_1_1_2_2"><i><font color="Green" title="E58">A18</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:42_1_1_2"/>
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="hahnban.html#E1:42_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">Y</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c13">f'</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c12">Y</font><b> such that </b><br/><a NAME="E10:42_1_1_2"/><i><font color="Green" title="E52">A19</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f'</font>
 <b>and </b><br/><a NAME="E11:42_1_1_2"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">Y</font> &amp; <font color="Maroon" title="c13">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c12">Y</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c13">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/><a NAME="E12:42_1_1_2"/><i><font color="Green" title="E53">A20</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">Y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E13:42_1_1_2"/>
<font color="Maroon" title="c11">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">u</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E14:42_1_1_2"/><b>then </b><i><font color="Green" title="E54">A21</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="hahnban.html#E10:42_1_1_2"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="hahnban.html#E12:42_1_1_2"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>set </b><font color="Maroon" title="c14">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span> ;<br/><a NAME="E15:42_1_1_2"/><i><font color="Green" title="E55">A22</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="hahnban.html#T5" target="_self" title="HAHNBAN:th.5">Th5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1_2_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c16">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E3:42_1_1_2_3"/><i><font color="Green" title="E56">A23</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 ;<br/><a NAME="E4:42_1_1_2_3"/>
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">Y</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c18">f'</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c17">Y</font><b> such that </b><br/><a NAME="E6:42_1_1_2_3"/><i><font color="Green" title="E57">A24</font></i>: 
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f'</font>
 <b>and </b><br/><a NAME="E7:42_1_1_2_3"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">Y</font> &amp; <font color="Maroon" title="c18">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c17">Y</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c18">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/><a NAME="E8:42_1_1_2_3"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">Y</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E6:42_1_1_2_3"><i><font color="Green" title="E57">A24</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E9:42_1_1_2_3"/>
<font color="Maroon" title="c15">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E3:42_1_1_2_3"><i><font color="Green" title="E56">A23</font></i></a>, <a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">D</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">u</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="hahnban.html#E14:42_1_1_2"><i><font color="Green" title="E54">A21</font></i></a>, <a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a>, <a class="ref" href="zfmisc_1.html#T94" title="ZFMISC_1:th.94">ZFMISC_1:94</a></i>;<br/><a NAME="E18:42_1_1_2"/>
<font color="Maroon" title="c15">D</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_4"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</a>
<div class="add"><b>let </b><font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c17">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/><b>assume </b><a NAME="E1:42_1_1_2_4_1"/><i><font color="Green" title="E56">A25</font></i>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font> &amp; <font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c18">D1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_4_1"/><i><font color="Green" title="E57">A26</font></i>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D1</font> &amp; <font color="Maroon" title="c18">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">g1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E5:42_1_1_2_4_1"/><i><font color="Green" title="E58">A27</font></i>: 
<font color="Maroon" title="c18">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">g1</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E3:42_1_1_2_4_1"><i><font color="Green" title="E57">A26</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">D2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:42_1_1_2_4_1"/><i><font color="Green" title="E59">A28</font></i>: 
( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">D2</font> &amp; <font color="Maroon" title="c20">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E1:42_1_1_2_4_1"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">g2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E9:42_1_1_2_4_1"/><i><font color="Green" title="E60">A29</font></i>: 
<font color="Maroon" title="c20">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c21">g2</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2_4_1"><i><font color="Green" title="E59">A28</font></i></a></i>;<br/><a NAME="E10:42_1_1_2_4_1"/>
( <font color="Maroon" title="c19">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E11:42_1_1_2_4_1"/><b>then </b>
( <font color="Maroon" title="c19">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">g2</font> or <font color="Maroon" title="c21">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">g1</font> )
 <b>by </b><i><a class="txt" href="hahnban.html#E2:42_1"><i><font color="Green" title="E48">A7</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E13:42_1_1_2_4_1"/><i><font color="Green" title="E61">A30</font></i>: 
( <font color="Maroon" title="c19">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">g</font> )
 ;<br/><a NAME="E14:42_1_1_2_4_1"/><i><font color="Green" title="E62">A31</font></i>: 
( <font color="Maroon" title="c18">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c20">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> )
 <b>by </b><i><a class="txt" href="hahnban.html#E5:42_1_1_2_4_1"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="hahnban.html#E9:42_1_1_2_4_1"><i><font color="Green" title="E60">A29</font></i></a>, <a class="txt" href="hahnban.html#E13:42_1_1_2_4_1"><i><font color="Green" title="E61">A30</font></i></a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/><a NAME="E15:42_1_1_2_4_1"/>
<font color="Maroon" title="c22">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">Y</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c24">f'</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c23">Y</font><b> such that </b><br/><a NAME="E17:42_1_1_2_4_1"/><i><font color="Green" title="E63">A32</font></i>: 
<font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f'</font>
 <b>and </b><br/><a NAME="E18:42_1_1_2_4_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c23">Y</font> &amp; <font color="Maroon" title="c24">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c23">Y</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c24">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/><a NAME="E19:42_1_1_2_4_1"/><i><font color="Green" title="E64">A33</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c23">Y</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E17:42_1_1_2_4_1"><i><font color="Green" title="E63">A32</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E20:42_1_1_2_4_1"/><b>then </b>
( <font color="Maroon" title="c16">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c23">Y</font> &amp; <font color="Maroon" title="c17">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c23">Y</font> )
 <b>by </b><i><a class="txt" href="hahnban.html#E3:42_1_1_2_4_1"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="hahnban.html#E7:42_1_1_2_4_1"><i><font color="Green" title="E59">A28</font></i></a>, <a class="txt" href="hahnban.html#E14:42_1_1_2_4_1"><i><font color="Green" title="E62">A31</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E21:42_1_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c16">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c23">Y</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T28" title="RLSUB_1:th.28">RLSUB_1:28</a></i>;<br/><a NAME="E22:42_1_1_2_4_1"/><b>then </b>
( <font color="Maroon" title="c16">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E19:42_1_1_2_4_1"><i><font color="Green" title="E64">A33</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>hence </b><a NAME="E23:42_1_1_2_4_1"/>
<font color="Maroon" title="c16">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c16">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c17">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E2:42_1_1_2_4"/>
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">D1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:42_1_1_2_4"/><i><font color="Green" title="E56">A34</font></i>: 
( <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D1</font> &amp; <font color="Maroon" title="c18">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E6:42_1_1_2_4"/><i><font color="Green" title="E57">A35</font></i>: 
<font color="Maroon" title="c18">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">g</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E4:42_1_1_2_4"><i><font color="Green" title="E56">A34</font></i></a></i>;<br/>
<a NAME="E7:42_1_1_2_4"/>
<font color="Maroon" title="c19">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">Y</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c21">f'</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c20">Y</font><b> such that </b><br/><a NAME="E9:42_1_1_2_4"/><i><font color="Green" title="E58">A36</font></i>: 
<font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f'</font>
 <b>and </b><br/><a NAME="E10:42_1_1_2_4"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">Y</font> &amp; <font color="Maroon" title="c21">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c20">Y</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c21">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/>
<a NAME="E11:42_1_1_2_4"/><i><font color="Green" title="E59">A37</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">Y</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E9:42_1_1_2_4"><i><font color="Green" title="E58">A36</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:42_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c17">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c20">Y</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E4:42_1_1_2_4"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="hahnban.html#E6:42_1_1_2_4"><i><font color="Green" title="E57">A35</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E13:42_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c17">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c20">Y</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T29" title="RLSUB_1:th.29">RLSUB_1:29</a></i>;<br/>
<a NAME="E14:42_1_1_2_4"/><b>then </b>
( <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 
<b>by </b><i><a class="txt" href="hahnban.html#E11:42_1_1_2_4"><i><font color="Green" title="E59">A37</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E15:42_1_1_2_4"/>
<font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c16">Y</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E20:42_1_1_2"/><i><font color="Green" title="E56">A38</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">u</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T43" title="RLSUB_1:th.43">RLSUB_1:43</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">u</font>;<br/><b>consider </b><font color="Maroon" title="c18">XX</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E23:42_1_1_2"/><i><font color="Green" title="E57">A39</font></i>: 
( <font color="Maroon" title="c17">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">XX</font> &amp; <font color="Maroon" title="c18">XX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">gg</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E25:42_1_1_2"/><i><font color="Green" title="E58">A40</font></i>: 
<font color="Maroon" title="c18">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">gg</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E23:42_1_1_2"><i><font color="Green" title="E57">A39</font></i></a></i>;<br/><a NAME="E26:42_1_1_2"/>
<font color="Maroon" title="c19">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">YY</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c21">ff</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c20">YY</font><b> such that </b><br/><a NAME="E28:42_1_1_2"/><i><font color="Green" title="E59">A41</font></i>: 
<font color="Maroon" title="c19">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">ff</font>
 <b>and </b><br/><a NAME="E29:42_1_1_2"/><i><font color="Green" title="E60">A42</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">YY</font>
 <b>and </b><br/><a NAME="E30:42_1_1_2"/><i><font color="Green" title="E61">A43</font></i>: 
<font color="Maroon" title="c21">ff</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>and </b><br/><a NAME="E31:42_1_1_2"/>
for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c20">YY</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c21">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/><a NAME="E32:42_1_1_2"/><i><font color="Green" title="E62">A44</font></i>: 
<font color="Maroon" title="c18">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">YY</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E25:42_1_1_2"><i><font color="Green" title="E58">A40</font></i></a>, <a class="txt" href="hahnban.html#E28:42_1_1_2"><i><font color="Green" title="E59">A41</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E33:42_1_1_2"/>
<font color="Maroon" title="c18">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">u</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E23:42_1_1_2"><i><font color="Green" title="E57">A39</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E34:42_1_1_2"/><b>then </b><i><font color="Green" title="E63">A45</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">Y</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="txt" href="hahnban.html#E29:42_1_1_2"><i><font color="Green" title="E60">A42</font></i></a>, <a class="txt" href="hahnban.html#E32:42_1_1_2"><i><font color="Green" title="E62">A44</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E35:42_1_1_2"/><i><font color="Green" title="E64">A46</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c21">ff</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E29:42_1_1_2"><i><font color="Green" title="E60">A42</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E36:42_1_1_2"/>
<font color="Maroon" title="c21">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">u</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E28:42_1_1_2"><i><font color="Green" title="E59">A41</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E37:42_1_1_2"/><b>then </b><i><font color="Green" title="E65">A47</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E30:42_1_1_2"><i><font color="Green" title="E61">A43</font></i></a>, <a class="txt" href="hahnban.html#E35:42_1_1_2"><i><font color="Green" title="E64">A46</font></i></a>, <a class="ref" href="grfunc_1.html#T88" title="GRFUNC_1:th.88">GRFUNC_1:88</a></i>;<br/><a NAME="E38:42_1_1_2"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="ref" href="partfun1.html#D5" title="PARTFUN1:def.5">PARTFUN1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">f'</font> = <font color="Maroon" title="c10">u</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">Y</font>, <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">f'</font> = <font color="Maroon" title="c22">f'</font> as   <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c16">Y</font> ;<br/><a NAME="E41:42_1_1_2"/><i><font color="Green" title="E66">A48</font></i>: 
<font color="Maroon" title="c23">f'</font> is <a href="hahnban.html#V2" title="HAHNBAN:attr.2">additive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">x</font>, <font color="Maroon" title="c25">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c16">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D4" target="_self" title="HAHNBAN:def.4">HAHNBAN:def 4</a><br/>

<b>consider </b><font color="Maroon" title="c26">XXx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:42_1_1_2_5"/><i><font color="Green" title="E67">A49</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">XXx</font> &amp; <font color="Maroon" title="c26">XXx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">ggx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E4:42_1_1_2_5"/><i><font color="Green" title="E68">A50</font></i>: 
<font color="Maroon" title="c26">XXx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">ggx</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E2:42_1_1_2_5"><i><font color="Green" title="E67">A49</font></i></a></i>;<br/>
<a NAME="E5:42_1_1_2_5"/><i><font color="Green" title="E69">A51</font></i>: 
<font color="Maroon" title="c27">ggx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">XXy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:42_1_1_2_5"/><i><font color="Green" title="E70">A52</font></i>: 
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">XXy</font> &amp; <font color="Maroon" title="c28">XXy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">ggy</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E9:42_1_1_2_5"/><i><font color="Green" title="E71">A53</font></i>: 
<font color="Maroon" title="c28">XXy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">ggy</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2_5"><i><font color="Green" title="E70">A52</font></i></a></i>;<br/>
<a NAME="E10:42_1_1_2_5"/>
<font color="Maroon" title="c29">ggy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:42_1_1_2_5"/><b>then </b>
( <font color="Maroon" title="c27">ggx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">ggy</font> or <font color="Maroon" title="c29">ggy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">ggx</font> )
 
<b>by </b><i><a class="txt" href="hahnban.html#E2:42_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="hahnban.html#E5:42_1_1_2_5"><i><font color="Green" title="E69">A51</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c30">gg</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E13:42_1_1_2_5"/><i><font color="Green" title="E72">A54</font></i>: 
( <font color="Maroon" title="c30">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">ggx</font> or <font color="Maroon" title="c30">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">ggy</font> )
 <b>and </b><br/><a NAME="E14:42_1_1_2_5"/><i><font color="Green" title="E73">A55</font></i>: 
( <font color="Maroon" title="c27">ggx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">gg</font> &amp; <font color="Maroon" title="c29">ggy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">gg</font> )
 ;<br/>
<a NAME="E15:42_1_1_2_5"/>
<font color="Maroon" title="c30">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c31">YY</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c32">ff</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c31">YY</font><b> such that </b><br/><a NAME="E17:42_1_1_2_5"/><i><font color="Green" title="E74">A56</font></i>: 
<font color="Maroon" title="c30">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">ff</font>
 <b>and </b><br/><a NAME="E18:42_1_1_2_5"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">YY</font> &amp; <font color="Maroon" title="c32">ff</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c31">YY</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c32">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/>
<a NAME="E19:42_1_1_2_5"/><i><font color="Green" title="E75">A57</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c32">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">YY</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:42_1_1_2_5"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">ggx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">gg</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">ggy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">gg</font> )
 
<b>by </b><i><a class="txt" href="hahnban.html#E14:42_1_1_2_5"><i><font color="Green" title="E73">A55</font></i></a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c33">x'</font> = <font color="Maroon" title="c24">x</font>, <font color="Maroon" title="c34">y'</font> = <font color="Maroon" title="c25">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c31">YY</font> <b>by </b><i><a class="txt" href="hahnban.html#E2:42_1_1_2_5"><i><font color="Green" title="E67">A49</font></i></a>, <a class="txt" href="hahnban.html#E4:42_1_1_2_5"><i><font color="Green" title="E68">A50</font></i></a>, <a class="txt" href="hahnban.html#E7:42_1_1_2_5"><i><font color="Green" title="E70">A52</font></i></a>, <a class="txt" href="hahnban.html#E9:42_1_1_2_5"><i><font color="Green" title="E71">A53</font></i></a>, <a class="txt" href="hahnban.html#E17:42_1_1_2_5"><i><font color="Green" title="E74">A56</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:42_1_1_2_5"/><i><font color="Green" title="E76">A58</font></i>: 
<font color="Maroon" title="c32">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">f'</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E17:42_1_1_2_5"><i><font color="Green" title="E74">A56</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E23:42_1_1_2_5"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">YY</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">Y</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="txt" href="hahnban.html#E2:42_1_1_2_5"><i><font color="Green" title="E67">A49</font></i></a>, <a class="txt" href="hahnban.html#E4:42_1_1_2_5"><i><font color="Green" title="E68">A50</font></i></a>, <a class="txt" href="hahnban.html#E7:42_1_1_2_5"><i><font color="Green" title="E70">A52</font></i></a>, <a class="txt" href="hahnban.html#E9:42_1_1_2_5"><i><font color="Green" title="E71">A53</font></i></a>, <a class="txt" href="hahnban.html#E13:42_1_1_2_5"><i><font color="Green" title="E72">A54</font></i></a>, <a class="txt" href="hahnban.html#E17:42_1_1_2_5"><i><font color="Green" title="E74">A56</font></i></a>, <a class="txt" href="hahnban.html#E19:42_1_1_2_5"><i><font color="Green" title="E75">A57</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E24:42_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c31">YY</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c16">Y</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T36" title="RLSUB_1:th.36">RLSUB_1:36</a></i>;<br/>
<a NAME="E25:42_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c24">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">x'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c34">y'</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T21" title="RLSUB_1:th.21">RLSUB_1:21</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c23">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c25">y</font></span>)</span> = 
<font color="Maroon" title="c32">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">x'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c34">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban.html#E19:42_1_1_2_5"><i><font color="Green" title="E75">A57</font></i></a>, <a class="txt" href="hahnban.html#E22:42_1_1_2_5"><i><font color="Green" title="E76">A58</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">x'</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#D4" target="_self" title="HAHNBAN:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban.html#E19:42_1_1_2_5"><i><font color="Green" title="E75">A57</font></i></a>, <a class="txt" href="hahnban.html#E22:42_1_1_2_5"><i><font color="Green" title="E76">A58</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban.html#E19:42_1_1_2_5"><i><font color="Green" title="E75">A57</font></i></a>, <a class="txt" href="hahnban.html#E22:42_1_1_2_5"><i><font color="Green" title="E76">A58</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E42:42_1_1_2"/>
<font color="Maroon" title="c23">f'</font> is <a href="hahnban.html#V3" title="HAHNBAN:attr.3">homogeneous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c16">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D5" target="_self" title="HAHNBAN:def.5">HAHNBAN:def 5</a><br/><b>let </b><font color="Maroon" title="c25">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>consider </b><font color="Maroon" title="c26">XX</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:42_1_1_2_6"/><i><font color="Green" title="E67">A59</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">XX</font> &amp; <font color="Maroon" title="c26">XX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">gg</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E4:42_1_1_2_6"/><i><font color="Green" title="E68">A60</font></i>: 
<font color="Maroon" title="c26">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">gg</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E2:42_1_1_2_6"><i><font color="Green" title="E67">A59</font></i></a></i>;<br/>
<a NAME="E5:42_1_1_2_6"/>
<font color="Maroon" title="c27">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c28">YY</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c29">ff</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c28">YY</font><b> such that </b><br/><a NAME="E7:42_1_1_2_6"/><i><font color="Green" title="E69">A61</font></i>: 
<font color="Maroon" title="c27">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">ff</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_6"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c28">YY</font>
 <b>and </b><br/><a NAME="E9:42_1_1_2_6"/>
<font color="Maroon" title="c29">ff</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>and </b><br/><a NAME="E10:42_1_1_2_6"/>
for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c28">YY</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c29">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/>
<a NAME="E11:42_1_1_2_6"/><i><font color="Green" title="E70">A62</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c28">YY</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">x'</font> = <font color="Maroon" title="c24">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c28">YY</font> <b>by </b><i><a class="txt" href="hahnban.html#E2:42_1_1_2_6"><i><font color="Green" title="E67">A59</font></i></a>, <a class="txt" href="hahnban.html#E4:42_1_1_2_6"><i><font color="Green" title="E68">A60</font></i></a>, <a class="txt" href="hahnban.html#E7:42_1_1_2_6"><i><font color="Green" title="E69">A61</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:42_1_1_2_6"/><i><font color="Green" title="E71">A63</font></i>: 
<font color="Maroon" title="c29">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">f'</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2_6"><i><font color="Green" title="E69">A61</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:42_1_1_2_6"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c28">YY</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">Y</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="txt" href="hahnban.html#E2:42_1_1_2_6"><i><font color="Green" title="E67">A59</font></i></a>, <a class="txt" href="hahnban.html#E4:42_1_1_2_6"><i><font color="Green" title="E68">A60</font></i></a>, <a class="txt" href="hahnban.html#E7:42_1_1_2_6"><i><font color="Green" title="E69">A61</font></i></a>, <a class="txt" href="hahnban.html#E11:42_1_1_2_6"><i><font color="Green" title="E70">A62</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:42_1_1_2_6"/><b>then </b>
<font color="Maroon" title="c28">YY</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c16">Y</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T36" title="RLSUB_1:th.36">RLSUB_1:36</a></i>;<br/>
<a NAME="E16:42_1_1_2_6"/><b>then </b>
<font color="Maroon" title="c25">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c30">x'</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T22" title="RLSUB_1:th.22">RLSUB_1:22</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c23">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c24">x</font></span>)</span> = 
<font color="Maroon" title="c29">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c30">x'</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban.html#E11:42_1_1_2_6"><i><font color="Green" title="E70">A62</font></i></a>, <a class="txt" href="hahnban.html#E13:42_1_1_2_6"><i><font color="Green" title="E71">A63</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c25">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#D5" target="_self" title="HAHNBAN:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c25">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban.html#E11:42_1_1_2_6"><i><font color="Green" title="E70">A62</font></i></a>, <a class="txt" href="hahnban.html#E13:42_1_1_2_6"><i><font color="Green" title="E71">A63</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
;<br/><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c24">f'</font> = <font color="Maroon" title="c23">f'</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c16">Y</font> <b>by </b><i><a class="txt" href="hahnban.html#E41:42_1_1_2"><i><font color="Green" title="E66">A48</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c16">Y</font>;<br/><b>let </b><font color="Maroon" title="c26">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>assume </b><a NAME="E1:42_1_1_2_7"/><i><font color="Green" title="E67">A64</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">v</font>
 ;<br/><b>consider </b><font color="Maroon" title="c27">XX</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_7"/><i><font color="Green" title="E68">A65</font></i>: 
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">XX</font> &amp; <font color="Maroon" title="c27">XX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="hahnban.html#E15:42_1_1_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="hahnban.html#E20:42_1_1_2"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c28">gg</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><b> such that </b><br/><a NAME="E5:42_1_1_2_7"/><i><font color="Green" title="E69">A66</font></i>: 
<font color="Maroon" title="c27">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c28">gg</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E3:42_1_1_2_7"><i><font color="Green" title="E68">A65</font></i></a></i>;<br/><a NAME="E6:42_1_1_2_7"/>
<font color="Maroon" title="c28">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c29">YY</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c30">ff</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c29">YY</font><b> such that </b><br/><a NAME="E8:42_1_1_2_7"/><i><font color="Green" title="E70">A67</font></i>: 
<font color="Maroon" title="c28">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">ff</font>
 <b>and </b><br/><a NAME="E9:42_1_1_2_7"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c29">YY</font>
 <b>and </b><br/><a NAME="E10:42_1_1_2_7"/>
<font color="Maroon" title="c30">ff</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>and </b><br/><a NAME="E11:42_1_1_2_7"/><i><font color="Green" title="E71">A68</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c29">YY</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c30">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E7:42_1_1_2"><i><font color="Green" title="E51">A11</font></i></a></i>;<br/><a NAME="E12:42_1_1_2_7"/><i><font color="Green" title="E72">A69</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c29">YY</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">y'</font> = <font color="Maroon" title="c25">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c29">YY</font> <b>by </b><i><a class="txt" href="hahnban.html#E3:42_1_1_2_7"><i><font color="Green" title="E68">A65</font></i></a>, <a class="txt" href="hahnban.html#E5:42_1_1_2_7"><i><font color="Green" title="E69">A66</font></i></a>, <a class="txt" href="hahnban.html#E8:42_1_1_2_7"><i><font color="Green" title="E70">A67</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E14:42_1_1_2_7"/><i><font color="Green" title="E73">A70</font></i>: 
<font color="Maroon" title="c30">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">f'</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E8:42_1_1_2_7"><i><font color="Green" title="E70">A67</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E15:42_1_1_2_7"/>
<font color="Maroon" title="c30">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">y'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:42_1_1_2_7"><i><font color="Green" title="E67">A64</font></i></a>, <a class="txt" href="hahnban.html#E11:42_1_1_2_7"><i><font color="Green" title="E71">A68</font></i></a></i>;<br/><b>hence </b><a NAME="E16:42_1_1_2_7"/>
<font color="Maroon" title="c24">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E12:42_1_1_2_7"><i><font color="Green" title="E72">A69</font></i></a>, <a class="txt" href="hahnban.html#E14:42_1_1_2_7"><i><font color="Green" title="E73">A70</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E45:42_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E34:42_1_1_2"><i><font color="Green" title="E63">A45</font></i></a>, <a class="txt" href="hahnban.html#E37:42_1_1_2"><i><font color="Green" title="E65">A47</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c25">u</font> = <font color="Maroon" title="c10">u</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font> ;<br/><b>take </b><font color="Maroon" title="c26">u</font> = <font color="Maroon" title="c25">u</font>;<br/><b>let </b><font color="Maroon" title="c27">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font>;<br/><b>assume </b><a NAME="E47:42_1_1_2"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ;<br/><b>hence </b><a NAME="E48:42_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c27">x</font>,<font color="Maroon" title="c26">u</font>]
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">psi</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font><b> such that </b><br/><a NAME="E14:42"/><i><font color="Green" title="E47">A71</font></i>: 
for <font color="Olive" title="b1">chi</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font>  st <font color="Maroon" title="c8">psi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">chi</font> holds <br/> not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">psi</font>,<font color="Olive" title="b1">chi</font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1" title="ORDERS_1:sch.1">ORDERS_1:sch 1</a>(<a class="txt" href="hahnban.html#E9:42"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="hahnban.html#E10:42"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="hahnban.html#E11:42"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="hahnban.html#E12:42"><i><font color="Green" title="E46">A5</font></i></a>);<br/></i>
<a NAME="E15:42"/>
<font color="Maroon" title="c8">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c9">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <b> such that </b><br/><a NAME="E17:42"/><i><font color="Green" title="E48">A72</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">psi</font>
 <b>and </b><br/><a NAME="E18:42"/><i><font color="Green" title="E49">A73</font></i>: 
 ex <font color="Olive" title="b1">Y</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font> &amp;  ex <font color="Olive" title="b2">f'</font> being  <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Olive" title="b1">Y</font> st <br/>( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f'</font> &amp; ( for <font color="Olive" title="b3">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Olive" title="b1">Y</font><br/> for <font color="Olive" title="b4">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> holds <br/><font color="Olive" title="b2">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">v</font> ) ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">Y</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E20:42"/><i><font color="Green" title="E50">A74</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font>
 <b>and </b><br/><a NAME="E21:42"/><i><font color="Green" title="E51">A75</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>and </b><br/><a NAME="E22:42"/><i><font color="Green" title="E52">A76</font></i>: 
 ex <font color="Olive" title="b1">f'</font> being  <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c10">Y</font> st <br/>( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f'</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c10">Y</font><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> holds <br/><font color="Olive" title="b1">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</font> ) )
 <b>by </b><i><a class="txt" href="hahnban.html#E18:42"><i><font color="Green" title="E49">A73</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">f'</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c10">Y</font><b> such that </b><br/><a NAME="E24:42"/><i><font color="Green" title="E53">A77</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font>
 <b>and </b><br/><a NAME="E25:42"/><i><font color="Green" title="E54">A78</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c10">Y</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c11">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E22:42"><i><font color="Green" title="E52">A76</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">RLSY</font> =  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c10">Y</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c10">Y</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c10">Y</font> #) as   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> <b>by </b><i><a class="txt" href="hahnban.html#E45"><i><font color="Green" title="E37">Lm2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">ggh</font> = <font color="Maroon" title="c11">f'</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c12">RLSY</font> <b>by </b><i><a class="txt" href="hahnban.html#E48"><i><font color="Green" title="E40">Lm5</font></i></a></i>;<br/>
<a NAME="E28:42"/><i><font color="Green" title="E55">A79</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ggh</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E24:42"><i><font color="Green" title="E53">A77</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">RLS</font> =  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> #) as   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> <b>by </b><i><a class="txt" href="hahnban.html#E45"><i><font color="Green" title="E37">Lm2</font></i></a></i>;<br/>
<a NAME="E30:42"/><i><font color="Green" title="E56">A80</font></i>: 
<font color="Maroon" title="c12">RLSY</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c14">RLS</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E46"><i><font color="Green" title="E38">Lm3</font></i></a></i>;<br/>
<div><i><font color="Green" title="E57">A81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_2"/>
 <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c10">Y</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c10">Y</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c10">Y</font> #) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> #)
 ;<br/><a NAME="E2:42_2"/><b>then </b>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> )
 <b>by </b><i><a class="txt" href="hahnban.html#E30:42"><i><font color="Green" title="E56">A80</font></i></a>, <a class="ref" href="rlsub_1.html#T40" title="RLSUB_1:th.40">RLSUB_1:40</a>, <a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/><a NAME="E3:42_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:42_2"/><i><font color="Green" title="E58">A82</font></i>: 
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; not <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c15">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="hahnban.html#E5:42_2"><i><font color="Green" title="E58">A82</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">phi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">Y</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:42_2"/><i><font color="Green" title="E59">A83</font></i>: 
<font color="Maroon" title="c17">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font>
 <b>and </b><br/><a NAME="E9:42_2"/><i><font color="Green" title="E60">A84</font></i>: 
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"><font color="Maroon" title="c10">Y</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c17">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E25:42"><i><font color="Green" title="E54">A78</font></i></a>, <a class="txt" href="hahnban.html#E5:42_2"><i><font color="Green" title="E58">A82</font></i></a>, <a class="txt" href="hahnban.html#E44"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/><a NAME="E10:42_2"/><i><font color="Green" title="E61">A85</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">Y</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:42_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">Y</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T14" target="_self" title="HAHNBAN:th.14">Th14</a></i>;<br/><a NAME="E12:42_2"/><b>then </b><i><font color="Green" title="E62">A86</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">Y</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E20:42"><i><font color="Green" title="E50">A74</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:42_2"/><i><font color="Green" title="E63">A87</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E10:42_2"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/><a NAME="E14:42_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">phi</font> = <font color="Maroon" title="c17">phi</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="txt" href="hahnban.html#E13:42_2"><i><font color="Green" title="E63">A87</font></i></a>, <a class="ref" href="partfun1.html#D5" title="PARTFUN1:def.5">PARTFUN1:def 5</a></i>;<br/><a NAME="E16:42_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E20:42"><i><font color="Green" title="E50">A74</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><a NAME="E17:42_2"/><b>then </b>
<font color="Maroon" title="c18">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E21:42"><i><font color="Green" title="E51">A75</font></i></a>, <a class="txt" href="hahnban.html#E24:42"><i><font color="Green" title="E53">A77</font></i></a>, <a class="txt" href="hahnban.html#E8:42_2"><i><font color="Green" title="E59">A83</font></i></a>, <a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></i>;<br/><a NAME="E18:42_2"/><b>then </b>
<font color="Maroon" title="c18">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E9:42_2"><i><font color="Green" title="E60">A84</font></i></a>, <a class="txt" href="hahnban.html#E12:42_2"><i><font color="Green" title="E62">A86</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">phi</font> = <font color="Maroon" title="c18">phi</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A</font> ;<br/><a NAME="E20:42_2"/><i><font color="Green" title="E64">A88</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:42_2"/><i><font color="Green" title="E65">A89</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c10">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T14" target="_self" title="HAHNBAN:th.14">Th14</a></i>;<br/><a NAME="E22:42_2"/>
<font color="Maroon" title="c16">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="rlvect_4.html#T12" title="RLVECT_4:th.12">RLVECT_4:12</a></i>;<br/><a NAME="E23:42_2"/><b>then </b>
<font color="Maroon" title="c16">v</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="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E24:42_2"/><b>then </b>
<font color="Maroon" title="c16">v</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"><span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c10">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E21:42_2"><i><font color="Green" title="E65">A89</font></i></a></i>;<br/><a NAME="E25:42_2"/><b>then </b><i><font color="Green" title="E66">A90</font></i>: 
<font color="Maroon" title="c19">phi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">psi</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E17:42"><i><font color="Green" title="E48">A72</font></i></a>, <a class="txt" href="hahnban.html#E24:42"><i><font color="Green" title="E53">A77</font></i></a>, <a class="txt" href="hahnban.html#E5:42_2"><i><font color="Green" title="E58">A82</font></i></a>, <a class="txt" href="hahnban.html#E10:42_2"><i><font color="Green" title="E61">A85</font></i></a>, <a class="txt" href="hahnban.html#E20:42_2"><i><font color="Green" title="E64">A88</font></i></a>, <a class="ref" href="rlsub_2.html#T9" title="RLSUB_2:th.9">RLSUB_2:9</a></i>;<br/><a NAME="E26:42_2"/>
<font color="Maroon" title="c8">psi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">phi</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E17:42"><i><font color="Green" title="E48">A72</font></i></a>, <a class="txt" href="hahnban.html#E24:42"><i><font color="Green" title="E53">A77</font></i></a>, <a class="txt" href="hahnban.html#E8:42_2"><i><font color="Green" title="E59">A83</font></i></a>, <a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/><b>hence </b><a NAME="E27:42_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hahnban.html#E14:42"><i><font color="Green" title="E47">A71</font></i></a>, <a class="txt" href="hahnban.html#E25:42_2"><i><font color="Green" title="E66">A90</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">psi</font> = <font color="Maroon" title="c8">psi</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="hahnban.html#E17:42"><i><font color="Green" title="E48">A72</font></i></a>, <a class="txt" href="hahnban.html#E28:42"><i><font color="Green" title="E55">A79</font></i></a>, <a class="txt" href="hahnban.html#E47"><i><font color="Green" title="E39">Lm4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">psi</font>
;<br/>

<b>thus </b><a NAME="E33:42"/>
<font color="Maroon" title="c15">psi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fi</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E17:42"><i><font color="Green" title="E48">A72</font></i></a>, <a class="txt" href="hahnban.html#E21:42"><i><font color="Green" title="E51">A75</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c16">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>thus </b><a NAME="E34:42"/>
<font color="Maroon" title="c15">psi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E17:42"><i><font color="Green" title="E48">A72</font></i></a>, <a class="txt" href="hahnban.html#E22:42"><i><font color="Green" title="E52">A76</font></i></a>, <a class="txt" href="hahnban.html#E31:42"><i><font color="Green" title="E57">A81</font></i></a></i>;<br/>


</div>
