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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="hahnban.html#NM2">Banach-Functional</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">fi</font> be   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><a NAME="E1:42"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a> ] <b>means </b> ex <font color="Olive">Y</font> being   <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">Y</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> &amp;  ex <font color="Olive">f'</font> being  <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Olive">Y</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f'</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Olive">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Olive">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) ) );<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a> </span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E3:42"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/>
<a NAME="E4:42"/><b>then </b>
( <font color="Maroon">fi</font> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="supinf_1.html#K3">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a>, <a class="ref" href="partfun1.html#T125">PARTFUN1:125</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fi'</font> = <font color="Maroon">fi</font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a>  ;<br/>
<a NAME="E6:42"/>
<font color="Maroon">fi'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T40">FUNCT_2:40</a></i>;<br/>
<a NAME="E7:42"/><b>then </b>
<font color="Maroon">fi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a> </span>)</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E9:42"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">x</font>]
 
;<br/>
<a NAME="E10:42"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:42"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green">E38</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">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:42_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:42_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">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">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">E41</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E3:42_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>hence </b><a NAME="E4:42_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">u</font>]
 <b>by </b><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">E42</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a> </span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">E</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/><a NAME="E4:42_1_1_2"/>
<font color="Maroon">B</font> is <a href="ordinal1.html#V6">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">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a><br/>

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

<b>hence </b><a NAME="E2:42_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">u</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">B</font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="hahnban.html#T13" target="_self">Th13</a>, </i>;<br/><b>consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><div><i><font color="Green">E112</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">t</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1_2_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">f</font> being    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_2"/><i><font color="Green">E114</font></i>: 
 ex <font color="Olive">Y</font> being   <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">Y</font> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> &amp;  ex <font color="Olive">f'</font> being  <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Olive">Y</font> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f'</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Olive">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Olive">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) ) )
 ;<br/><b>consider </b><font color="Maroon">Y</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:42_1_1_2_2"/><i><font color="Green">E115</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E7:42_1_1_2_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_2"/><i><font color="Green">E120</font></i>: 
 ex <font color="Olive">f'</font> being  <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f'</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Olive">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) )
 <b>by </b><i><a class="ref" href="hahnban.html#T14" target="_self">Th14</a></i>;<br/><b>consider </b><font color="Maroon">f'</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E10:42_1_1_2_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>and </b><br/><a NAME="E11:42_1_1_2_2"/><i><font color="Green">E122</font></i>: 
for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f'</font> as   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font> ;<br/><b>take </b><font color="Maroon">f</font> = <font color="Maroon">f</font>;<br/><b>thus </b><a NAME="E13:42_1_1_2_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E14:42_1_1_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E15:42_1_1_2_2"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E16:42_1_1_2_2"/>
for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E8:42_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">f'</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E10:42_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>and </b><br/><a NAME="E11:42_1_1_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) )
 <b>by </b><i/>;<br/><a NAME="E12:42_1_1_2"/><i><font color="Green">E127</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E13:42_1_1_2"/>
<font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E14:42_1_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="hahnban.html#T14" target="_self">Th14</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>set </b><font color="Maroon">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span> ;<br/><a NAME="E15:42_1_1_2"/><i><font color="Green">E129</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span> 
 <b>by </b><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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E3:42_1_1_2_3"/><i><font color="Green">E130</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E4:42_1_1_2_3"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">f'</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E6:42_1_1_2_3"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>and </b><br/><a NAME="E7:42_1_1_2_3"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) )
 <b>by </b><i/>;<br/><a NAME="E8:42_1_1_2_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E9:42_1_1_2_3"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">D</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i>, , <a class="ref" href="relat_1.html#T64">RELAT_1:64</a>, <a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/><a NAME="E18:42_1_1_2"/>
<font color="Maroon">D</font> is <a href="rlsub_1.html#V1">lineary-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">RLSUB_1:def 1</a>
<div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:42_1_1_2_4_1"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 ;<br/><b>then consider </b><font color="Maroon">D1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_4_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">g1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:42_1_1_2_4_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">D2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:42_1_1_2_4_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">g2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E9:42_1_1_2_4_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">D2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E10:42_1_1_2_4_1"/>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E11:42_1_1_2_4_1"/><b>then </b>
( <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g2</font> or <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g1</font> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E13:42_1_1_2_4_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> )
 ;<br/><a NAME="E14:42_1_1_2_4_1"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , , <a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/><a NAME="E15:42_1_1_2_4_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">f'</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E17:42_1_1_2_4_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>and </b><br/><a NAME="E18:42_1_1_2_4_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) )
 <b>by </b><i/>;<br/><a NAME="E19:42_1_1_2_4_1"/><i><font color="Green">E140</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T17" target="_self">Th17</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E20:42_1_1_2_4_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="hahnban.html#T15" target="_self">Th15</a>, <a class="ref" href="hahnban.html#T16" target="_self">Th16</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E21:42_1_1_2_4_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T28">RLSUB_1:28</a></i>;<br/><a NAME="E22:42_1_1_2_4_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="hahnban.html#T18" target="_self">Th18</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E23:42_1_1_2_4_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>



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

<b>then consider </b><font color="Maroon">D1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:42_1_1_2_4"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E6:42_1_1_2_4"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:42_1_1_2_4"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">f'</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E9:42_1_1_2_4"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>and </b><br/><a NAME="E10:42_1_1_2_4"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E11:42_1_1_2_4"/><i><font color="Green">E144</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:42_1_1_2_4"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E13:42_1_1_2_4"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T29">RLSUB_1:29</a></i>;<br/>
<a NAME="E14:42_1_1_2_4"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 
<b>by </b><i><a class="ref" href="hahnban.html#T15" target="_self">Th15</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:42_1_1_2_4"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">Y</font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E20:42_1_1_2"/><i><font color="Green">E145</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T43">RLSUB_1:43</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font>;<br/><b>consider </b><font color="Maroon">XX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:42_1_1_2"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">XX</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">gg</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E25:42_1_1_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">XX</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font>
 <b>by </b><i/>;<br/><a NAME="E26:42_1_1_2"/>
<font color="Maroon">gg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">YY</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">ff</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">YY</font><b> such that </b><br/><a NAME="E28:42_1_1_2"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font>
 <b>and </b><br/><a NAME="E29:42_1_1_2"/><i><font color="Green">E172</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font>
 <b>and </b><br/><a NAME="E30:42_1_1_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>and </b><br/><a NAME="E31:42_1_1_2"/>
for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">YY</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 <b>by </b><i/>;<br/><a NAME="E32:42_1_1_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">XX</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font>
 <b>by </b><i>, <a class="ref" href="hahnban.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E33:42_1_1_2"/>
<font color="Maroon">XX</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E34:42_1_1_2"/><b>then </b><i><font color="Green">E175</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="hahnban.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E35:42_1_1_2"/><i><font color="Green">E176</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E36:42_1_1_2"/>
<font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T15" target="_self">Th15</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E37:42_1_1_2"/><b>then </b><i><font color="Green">E177</font></i>: 
<font color="Maroon">u</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>by </b><i>, <a class="ref" href="hahnban.html#T18" target="_self">Th18</a>, <a class="ref" href="grfunc_1.html#T88">GRFUNC_1:88</a></i>;<br/><a NAME="E38:42_1_1_2"/>
 ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">u</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>, <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f'</font> as   <a href="hahnban.html#NM1">Functional</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E41:42_1_1_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">f'</font> is <a href="hahnban.html#V2" target="_self">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">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D4" target="_self">HAHNBAN:def 4</a><br/>

<b>consider </b><font color="Maroon">XXx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:42_1_1_2_5"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XXx</font> &amp; <font color="Maroon">XXx</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">ggx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E4:42_1_1_2_5"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">XXx</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ggx</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E5:42_1_1_2_5"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">ggx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">XXy</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:42_1_1_2_5"/><i><font color="Green">E185</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XXy</font> &amp; <font color="Maroon">XXy</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">ggy</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E9:42_1_1_2_5"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">XXy</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ggy</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:42_1_1_2_5"/>
<font color="Maroon">ggy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:42_1_1_2_5"/><b>then </b>
( <font color="Maroon">ggx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ggy</font> or <font color="Maroon">ggy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ggx</font> )
 
<b>by </b><i>, <a class="ref" href="hahnban.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">gg</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E13:42_1_1_2_5"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ggx</font> or <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ggy</font> )
 <b>and </b><br/><a NAME="E14:42_1_1_2_5"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">ggx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">gg</font> &amp; <font color="Maroon">ggy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">gg</font> )
 ;<br/>
<a NAME="E15:42_1_1_2_5"/>
<font color="Maroon">gg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">YY</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">ff</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">YY</font><b> such that </b><br/><a NAME="E17:42_1_1_2_5"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font>
 <b>and </b><br/><a NAME="E18:42_1_1_2_5"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font> &amp; <font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">YY</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E19:42_1_1_2_5"/><i><font color="Green">E191</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:42_1_1_2_5"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ggx</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ggy</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">YY</font> <b>by </b><i><a class="ref" href="hahnban.html#T21" target="_self">Th21</a>, , , , , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:42_1_1_2_5"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f'</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E23:42_1_1_2_5"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="hahnban.html#T21" target="_self">Th21</a>, , , , , , <a class="ref" href="hahnban.html#T23" target="_self">Th23</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E24:42_1_1_2_5"/><b>then </b>
<font color="Maroon">YY</font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T36">RLSUB_1:36</a></i>;<br/>
<a NAME="E25:42_1_1_2_5"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T21">RLSUB_1:21</a></i>;<br/>
<b>hence </b><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T23" target="_self">Th23</a>, <a class="ref" href="hahnban.html#T24" target="_self">Th24</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T23" target="_self">Th23</a>, <a class="ref" href="hahnban.html#T24" target="_self">Th24</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T23" target="_self">Th23</a>, <a class="ref" href="hahnban.html#T24" target="_self">Th24</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E42:42_1_1_2"/>
<font color="Maroon">f'</font> is <a href="hahnban.html#V3" target="_self">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">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D5" target="_self">HAHNBAN:def 5</a><br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>consider </b><font color="Maroon">XX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:42_1_1_2_6"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">XX</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">gg</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E4:42_1_1_2_6"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">XX</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E5:42_1_1_2_6"/>
<font color="Maroon">gg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">YY</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">ff</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">YY</font><b> such that </b><br/><a NAME="E7:42_1_1_2_6"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_6"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font>
 <b>and </b><br/><a NAME="E9:42_1_1_2_6"/>
<font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>and </b><br/><a NAME="E10:42_1_1_2_6"/>
for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">YY</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:42_1_1_2_6"/><i><font color="Green">E196</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">YY</font> <b>by </b><i><a class="ref" href="hahnban.html#T21" target="_self">Th21</a>, , <a class="ref" href="hahnban.html#T22" target="_self">Th22</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:42_1_1_2_6"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f'</font>
 
<b>by </b><i><a class="ref" href="hahnban.html#T22" target="_self">Th22</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:42_1_1_2_6"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="hahnban.html#T21" target="_self">Th21</a>, , <a class="ref" href="hahnban.html#T22" target="_self">Th22</a>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:42_1_1_2_6"/><b>then </b>
<font color="Maroon">YY</font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T36">RLSUB_1:36</a></i>;<br/>
<a NAME="E16:42_1_1_2_6"/><b>then </b>
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/>
<b>hence </b><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f'</font> as   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="hahnban.html#T20" target="_self">Th20</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">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:42_1_1_2_7"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><b>consider </b><font color="Maroon">XX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_7"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">XX</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> : verum </span>}</span>  )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">gg</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:42_1_1_2_7"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">XX</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font>
 <b>by </b><i/>;<br/><a NAME="E6:42_1_1_2_7"/>
<font color="Maroon">gg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">YY</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">ff</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">YY</font><b> such that </b><br/><a NAME="E8:42_1_1_2_7"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font>
 <b>and </b><br/><a NAME="E9:42_1_1_2_7"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font>
 <b>and </b><br/><a NAME="E10:42_1_1_2_7"/>
<font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>and </b><br/><a NAME="E11:42_1_1_2_7"/><i><font color="Green">E202</font></i>: 
for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">YY</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 <b>by </b><i/>;<br/><a NAME="E12:42_1_1_2_7"/><i><font color="Green">E203</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">YY</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">YY</font> <b>by </b><i>, <a class="ref" href="hahnban.html#T22" target="_self">Th22</a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E14:42_1_1_2_7"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f'</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E15:42_1_1_2_7"/>
<font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T21" target="_self">Th21</a>, </i>;<br/><b>hence </b><a NAME="E16:42_1_1_2_7"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E45:42_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T17" target="_self">Th17</a>, <a class="ref" href="hahnban.html#T19" target="_self">Th19</a></i>;<br/><b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> ;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E47:42_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>hence </b><a NAME="E48:42_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">u</font>]
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">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">psi</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E14:42"/><i><font color="Green">E205</font></i>: 
for <font color="Olive">chi</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>  st <font color="Maroon">psi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">chi</font> holds <br/> not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">psi</font>,<font color="Olive">chi</font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1">ORDERS_1:sch 1</a>(, <a class="ref" href="hahnban.html#T8" target="_self">Th8</a>, , <a class="ref" href="hahnban.html#T9" target="_self">Th9</a>);<br/></i>
<a NAME="E15:42"/>
<font color="Maroon">psi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon">f</font> being    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E17:42"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font>
 <b>and </b><br/><a NAME="E18:42"/><i><font color="Green">E208</font></i>: 
 ex <font color="Olive">Y</font> being   <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">Y</font> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> &amp;  ex <font color="Olive">f'</font> being  <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Olive">Y</font> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f'</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Olive">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Olive">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) ) )
 ;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E20:42"/><i><font color="Green">E209</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E21:42"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>and </b><br/><a NAME="E22:42"/><i><font color="Green">E211</font></i>: 
 ex <font color="Olive">f'</font> being  <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f'</font> &amp; ( for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Olive">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f'</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E24:42"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>and </b><br/><a NAME="E25:42"/><i><font color="Green">E213</font></i>: 
for <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">RLSY</font> =  <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">Y</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">Y</font> #) as   <a href="rlvect_1.html#NM2">RealLinearSpace</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ggh</font> = <font color="Maroon">f'</font> as   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">RLSY</font> <b>by </b><i/>;<br/>
<a NAME="E28:42"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ggh</font>
 
<b>by </b><i><a class="ref" href="hahnban.html#T14" target="_self">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">RLS</font> =  <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> #) as   <a href="rlvect_1.html#NM2">RealLinearSpace</a> <b>by </b><i/>;<br/>
<a NAME="E30:42"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">RLSY</font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">RLS</font>
 
<b>by </b><i><a class="txt" href="hahnban.html#E12:42"><i><font color="Green">E38</font></i></a></i>;<br/>
<div><i><font color="Green">E219</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">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">Y</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">Y</font> #) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> #)
 ;<br/><a NAME="E2:42_2"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T40">RLSUB_1:40</a>, <a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><a NAME="E3:42_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="xboole_0.html#R2">c&lt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:42_2"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="rlsub_2.html#T77">RLSUB_2:77</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">phi</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:42_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">phi</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>and </b><br/><a NAME="E9:42_2"/><i><font color="Green">E223</font></i>: 
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">phi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 <b>by </b><i>, , <a class="txt" href="hahnban.html#E11:42"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E10:42_2"/><i><font color="Green">E224</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:42_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E12:42_2"/><b>then </b><i><font color="Green">E225</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:42_2"/><i><font color="Green">E226</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><a NAME="E14:42_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">phi</font> = <font color="Maroon">phi</font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="hahnban.html#T17" target="_self">Th17</a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><a NAME="E16:42_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E17:42_2"/><b>then </b>
<font color="Maroon">phi</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>by </b><i>, <a class="ref" href="hahnban.html#T14" target="_self">Th14</a>, <a class="ref" href="hahnban.html#T15" target="_self">Th15</a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>;<br/><a NAME="E18:42_2"/><b>then </b>
<font color="Maroon">phi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="hahnban.html#T16" target="_self">Th16</a></i>;<br/><b>then reconsider </b><font color="Maroon">phi</font> = <font color="Maroon">phi</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E20:42_2"/><i><font color="Green">E227</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:42_2"/><i><font color="Green">E228</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="rlsub_2.html#K1">+</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E22:42_2"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rlvect_4.html#T12">RLVECT_4:12</a></i>;<br/><a NAME="E23:42_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E24:42_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="rlsub_2.html#K1">+</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E25:42_2"/><b>then </b><i><font color="Green">E229</font></i>: 
<font color="Maroon">phi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">psi</font>
 <b>by </b><i>, <a class="ref" href="hahnban.html#T14" target="_self">Th14</a>, , , <a class="ref" href="hahnban.html#T18" target="_self">Th18</a>, <a class="ref" href="rlsub_2.html#T9">RLSUB_2:9</a></i>;<br/><a NAME="E26:42_2"/>
<font color="Maroon">psi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">phi</font>
 <b>by </b><i>, <a class="ref" href="hahnban.html#T14" target="_self">Th14</a>, <a class="ref" href="hahnban.html#T15" target="_self">Th15</a>, <a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/><b>hence </b><a NAME="E27:42_2"/>
contradiction
 <b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="hahnban.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">psi</font> = <font color="Maroon">psi</font> as   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">V</font> <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">psi</font>
;<br/>

<b>thus </b><a NAME="E33:42"/>
<font color="Maroon">psi</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>

<b>thus </b><a NAME="E34:42"/>
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div>
