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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">q</font> being   <a href="hahnban.html#NM2" title="HAHNBAN:NM.2">Banach-Functional</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b3">fi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Olive" title="b1">X</font>  st (  for <font color="Olive" title="b4">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b5">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="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> holds <br/><font color="Olive" title="b3">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> ) holds <br/> ex <font color="Olive" title="b4">psi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b4">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="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">fi</font> &amp; (  for <font color="Olive" title="b5">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b4">psi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">x</font> ) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being   <a href="hahnban.html#NM2" title="HAHNBAN:NM.2">Banach-Functional</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">fi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c2">X</font>  st (  for <font color="Olive" title="b3">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="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">x</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">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">v</font> ) holds <br/> ex <font color="Olive" title="b3">psi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b3">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="Olive" title="b2">fi</font> &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b3">psi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> ) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">fi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c2">X</font>  st (  for <font color="Olive" title="b2">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="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">x</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">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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="b3">v</font> ) holds <br/> ex <font color="Olive" title="b2">psi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">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="Olive" title="b1">fi</font> &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b2">psi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">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="b3">x</font> ) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="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> ) implies  ex <font color="Olive" title="b1">psi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">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> &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b1">psi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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">x</font> ) ) )</span><br/>







<a NAME="E1:32"/>
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></span>;<br/>
<a NAME="E2:32"/><span class="kw">then </span>
( <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="numbers.html#K1" title="NUMBERS:func.1">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="numbers.html#K1" title="NUMBERS:func.1">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" title="PARTFUN1:th.45">PARTFUN1:45</a>, <a class="ref" href="partfun1.html#T50" title="PARTFUN1:th.50">PARTFUN1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">fi9</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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">RLS</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">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">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> #) as   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="hahnban.html#E30"><span class="lab"><font color="Green" title="E30">Lm2</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)] <span class="kw">means </span> 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">f9</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">f9</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">f9</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">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="numbers.html#K1" title="NUMBERS:func.1">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span> <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<a NAME="E6:32"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c5">fi9</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T19" title="RELSET_1:th.19">RELSET_1:19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:32"/>
 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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">psi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">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> &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b1">psi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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">x</font> ) )</span><br/>

<a NAME="E8:32"/><span class="kw">then </span>
<font color="Maroon" title="c4">fi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E35">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c7">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $2;<br/>
<a NAME="E10:32"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
 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="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E11:32"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
 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="c8">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>]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E38">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> &amp; (  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="c8">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="c9">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="c9">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>] ) implies  ex <font color="Olive" title="b1">u</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font> st <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive" title="b2">x</font>] )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">and </span><br/><a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
 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="c8">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="c9">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="c9">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>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font> st <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive" title="b2">x</font>]</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1"/>
( <font color="Maroon" title="c9">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="c9">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="32_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font> st <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive" title="b2">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">u</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font>;<br/><span class="kw">take </span><font color="Maroon" title="c11">u</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">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="c9">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">u</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">u</font>] )</span><br/><span class="kw">assume </span><a NAME="E2:32_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">u</font>]</span><br/><span class="kw">hence </span><a NAME="E3:32_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">u</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c9">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font> st <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive" title="b2">x</font>]</span><br/><div class="add"><a NAME="E2:32_1_1_2"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c9">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="32_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> or  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> or <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_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="c9">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="c9">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:32_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:32_1_1_2"/>
<font color="Maroon" title="c9">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">u</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="trees_2.html#T40" title="TREES_2:th.40">TREES_2:40</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">E</font> = <font color="Maroon" title="c9">B</font> as   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E1:32_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">t</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">B</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">K</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> ;<br/><a NAME="E6:32_1_1_2"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T110" title="FUNCT_1:th.110">FUNCT_1:110</a></span>;<br/><div><span class="lab"><font color="Green" title="E44">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> implies  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> 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="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp;  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="Olive" title="b2">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="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> ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:32_1_1_2_2"/>
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> 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="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp;  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="Olive" title="b2">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="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> ) )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)<span class="kw"> such that </span><br/><a NAME="E3:32_1_1_2_2"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_2_2"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
 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="c15">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">f9</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="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f9</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">f9</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/><span class="kw">consider </span><font color="Maroon" title="c16">Y</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E6:32_1_1_2_2"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
 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>
 <span class="kw">and </span><br/><a NAME="E7:32_1_1_2_2"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
<font color="Maroon" title="c15">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>
 <span class="kw">and </span><br/><a NAME="E8:32_1_1_2_2"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
 ex <font color="Olive" title="b1">f9</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c16">Y</font> st <br/>( <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f9</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="c16">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">f9</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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_2_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c17">Y</font> = <font color="Maroon" title="c16">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c17">Y</font> st <br/>( <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  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="Olive" title="b1">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="b2">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="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> ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c18">f9</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c17">Y</font><span class="kw"> such that </span><br/><a NAME="E10:32_1_1_2_2"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
<font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f9</font>
 <span class="kw">and </span><br/><a NAME="E11:32_1_1_2_2"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
 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">f9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1_2_2"><span class="lab"><font color="Green" title="E49">A16</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">f</font> = <font color="Maroon" title="c18">f9</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c17">Y</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c20">f</font> = <font color="Maroon" title="c19">f</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f</font> &amp;  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="c20">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="c20">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> ) )</span><br/><span class="kw">thus </span><a NAME="E13:32_1_1_2_2"/>
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_2_2"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E10:32_1_1_2_2"><span class="lab"><font color="Green" title="E50">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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="c20">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="c20">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> ) )</span><br/><span class="kw">thus </span><a NAME="E14:32_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="c17">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">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="c20">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> ) )</span><br/><span class="kw">thus </span><a NAME="E15:32_1_1_2_2"/>
<font color="Maroon" title="c20">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E10:32_1_1_2_2"><span class="lab"><font color="Green" title="E50">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being   <a href="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="c20">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></span><br/><span class="kw">thus </span><a NAME="E16:32_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="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="c20">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_1_1_2_2"><span class="lab"><font color="Green" title="E51">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E45">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span>  implies <font color="Maroon" title="c14">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> )</span><br/><span class="kw">assume </span><a NAME="E1:32_1_1_2_3"/>
<font color="Maroon" title="c14">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">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></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_2_3"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 ;<br/><a NAME="E4:32_1_1_2_3"/>
<font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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="c17">f9</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c16">Y</font><span class="kw"> such that </span><br/><a NAME="E6:32_1_1_2_3"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f9</font>
 <span class="kw">and </span><br/><a NAME="E7:32_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="c16">Y</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1_1_2_3"/>
<font color="Maroon" title="c17">f9</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>
 <span class="kw">and </span><br/><a NAME="E9:32_1_1_2_3"/>
 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="c17">f9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>;<br/><a NAME="E10:32_1_1_2_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">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="c16">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2_3"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:32_1_1_2_3"/>
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_2_3"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:32_1_1_2"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E1:32_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E10:32_1_1_2"/><span class="kw">then </span>
 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> ex <font color="Olive" title="b2">f9</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Olive" title="b1">Y</font> st <br/>(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f9</font> &amp;  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="Olive" title="b2">f9</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="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">f9</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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>;<br/><a NAME="E11:32_1_1_2"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">D</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E8:32_1_1_2"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T76" title="ZFMISC_1:th.76">ZFMISC_1:76</a></span>;<br/><a NAME="E13:32_1_1_2"/>
<font color="Maroon" title="c14">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="32_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> or <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c15">v</font>, <font color="Maroon" title="c16">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> &amp; <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> implies <font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 <span class="kw">and </span><br/><a NAME="E2:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c17">D1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
<font color="Maroon" title="c17">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E1:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">g1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E7:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<font color="Maroon" title="c17">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">D2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">D2</font>
 <span class="kw">and </span><br/><a NAME="E10:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<font color="Maroon" title="c19">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E2:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">g2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E12:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<font color="Maroon" title="c19">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a></span>;<br/><a NAME="E13:32_1_1_2_4_1"/>
( <font color="Maroon" title="c18">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E14:32_1_1_2_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c18">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">g2</font> or <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">g1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E16:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
<font color="Maroon" title="c18">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">and </span><br/><a NAME="E17:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E55">A31</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">g</font>
 ;<br/><a NAME="E18:32_1_1_2_4_1"/>
<font color="Maroon" title="c21">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">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="c23">f9</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c22">Y</font><span class="kw"> such that </span><br/><a NAME="E20:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E56">A32</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f9</font>
 <span class="kw">and </span><br/><a NAME="E21:32_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="c22">Y</font>
 <span class="kw">and </span><br/><a NAME="E22:32_1_1_2_4_1"/>
<font color="Maroon" title="c23">f9</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>
 <span class="kw">and </span><br/><a NAME="E23:32_1_1_2_4_1"/>
 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="c22">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="c23">f9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>;<br/><a NAME="E24:32_1_1_2_4_1"/><span class="lab"><font color="Green" title="E57">A33</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">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="c22">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E25:32_1_1_2_4_1"/>
<font color="Maroon" title="c19">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E17:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="ref" href="relat_1.html#T11" title="RELAT_1:th.11">RELAT_1:11</a></span>;<br/><a NAME="E26:32_1_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A34</font></span>: 
<font color="Maroon" title="c16">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E24:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E27:32_1_1_2_4_1"/>
<font color="Maroon" title="c17">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E16:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="relat_1.html#T11" title="RELAT_1:th.11">RELAT_1:11</a></span>;<br/><a NAME="E28:32_1_1_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E24:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E29:32_1_1_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="rlsub_1.html#T20" title="RLSUB_1:th.20">RLSUB_1:20</a></span>;<br/><a NAME="E30:32_1_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E31:32_1_1_2_4_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E32:32_1_1_2_4_1"/>
<font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E30:32_1_1_2_4_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c15">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> or <font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> or <font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:32_1_1_2_4"/>
<font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">D1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:32_1_1_2_4"/><span class="lab"><font color="Green" title="E46">A36</font></span>: 
<font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">D1</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_2_4"/><span class="lab"><font color="Green" title="E47">A37</font></span>: 
<font color="Maroon" title="c17">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E7:32_1_1_2_4"/><span class="lab"><font color="Green" title="E48">A38</font></span>: 
<font color="Maroon" title="c17">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_2_4"><span class="lab"><font color="Green" title="E47">A37</font></span></a></span>;<br/>
<a NAME="E8:32_1_1_2_4"/>
<font color="Maroon" title="c18">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">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="c20">f9</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c19">Y</font><span class="kw"> such that </span><br/><a NAME="E10:32_1_1_2_4"/><span class="lab"><font color="Green" title="E49">A39</font></span>: 
<font color="Maroon" title="c18">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f9</font>
 <span class="kw">and </span><br/><a NAME="E11:32_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="c19">Y</font>
 <span class="kw">and </span><br/><a NAME="E12:32_1_1_2_4"/>
<font color="Maroon" title="c20">f9</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>
 <span class="kw">and </span><br/><a NAME="E13:32_1_1_2_4"/>
 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="c19">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="c20">f9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>;<br/>
<a NAME="E14:32_1_1_2_4"/><span class="lab"><font color="Green" title="E50">A40</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">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="c19">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1_2_4"><span class="lab"><font color="Green" title="E49">A39</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:32_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c16">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c19">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_2_4"><span class="lab"><font color="Green" title="E46">A36</font></span></a>, <a class="txt" href="#E7:32_1_1_2_4"><span class="lab"><font color="Green" title="E48">A38</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E16:32_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c19">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T21" title="RLSUB_1:th.21">RLSUB_1:21</a></span>;<br/>
<a NAME="E17:32_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A41</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_1_1_2_4"><span class="lab"><font color="Green" title="E50">A40</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E18:32_1_1_2_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E19:32_1_1_2_4"/>
<font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E17:32_1_1_2_4"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">Y</font> being   <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E15:32_1_1_2"/><span class="lab"><font color="Green" title="E46">A42</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T35" title="RLSUB_1:th.35">RLSUB_1:35</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">t</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">u</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">XX</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:32_1_1_2"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">XX</font>
 <span class="kw">and </span><br/><a NAME="E18:32_1_1_2"/><span class="lab"><font color="Green" title="E47">A43</font></span>: 
<font color="Maroon" title="c17">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E19:32_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#NK2" title="RELAT_1:NK.2">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D3" title="PARTFUN1:def.3">PARTFUN1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">f9</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="c15">Y</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></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#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">f9</font> = <font color="Maroon" title="c18">f9</font> as   <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c15">Y</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c20">gg</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E23:32_1_1_2"/><span class="lab"><font color="Green" title="E48">A44</font></span>: 
<font color="Maroon" title="c17">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">gg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32_1_1_2"><span class="lab"><font color="Green" title="E47">A43</font></span></a></span>;<br/><a NAME="E24:32_1_1_2"/><span class="lab"><font color="Green" title="E49">A45</font></span>: 
<font color="Maroon" title="c19">f9</font> is  <a href="hahnban.html#V2" title="HAHNBAN:attr.2">additive</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font>, <font color="Maroon" title="c22">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c15">Y</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="hahnban.html#D2" target="_self" title="HAHNBAN:def.2">HAHNBAN:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c22">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">y</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c23">XXx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:32_1_1_2_5"/><span class="lab"><font color="Green" title="E50">A46</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">XXx</font>
 <span class="kw">and </span><br/><a NAME="E3:32_1_1_2_5"/><span class="lab"><font color="Green" title="E51">A47</font></span>: 
<font color="Maroon" title="c23">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">ggx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E5:32_1_1_2_5"/><span class="lab"><font color="Green" title="E52">A48</font></span>: 
<font color="Maroon" title="c23">XXx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">ggx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_2_5"><span class="lab"><font color="Green" title="E51">A47</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">XXy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:32_1_1_2_5"/><span class="lab"><font color="Green" title="E53">A49</font></span>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">XXy</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1_1_2_5"/><span class="lab"><font color="Green" title="E54">A50</font></span>: 
<font color="Maroon" title="c25">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">ggy</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E10:32_1_1_2_5"/><span class="lab"><font color="Green" title="E55">A51</font></span>: 
<font color="Maroon" title="c25">XXy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">ggy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1_2_5"><span class="lab"><font color="Green" title="E54">A50</font></span></a></span>;<br/>
<a NAME="E11:32_1_1_2_5"/>
( <font color="Maroon" title="c24">ggx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c26">ggy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E12:32_1_1_2_5"/><span class="kw">then </span>
( <font color="Maroon" title="c24">ggx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">ggy</font> or <font color="Maroon" title="c26">ggy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">ggx</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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="c11">E</font><span class="kw"> such that </span><br/><a NAME="E14:32_1_1_2_5"/><span class="lab"><font color="Green" title="E56">A52</font></span>: 
( <font color="Maroon" title="c27">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">ggx</font> or <font color="Maroon" title="c27">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">ggy</font> )
 <span class="kw">and </span><br/><a NAME="E15:32_1_1_2_5"/><span class="lab"><font color="Green" title="E57">A53</font></span>: 
( <font color="Maroon" title="c24">ggx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">gg</font> &amp; <font color="Maroon" title="c26">ggy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">gg</font> )
 ;<br/>
<a NAME="E16:32_1_1_2_5"/>
<font color="Maroon" title="c27">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E18:32_1_1_2_5"/><span class="lab"><font color="Green" title="E58">A54</font></span>: 
<font color="Maroon" title="c27">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">ff</font>
 <span class="kw">and </span><br/><a NAME="E19:32_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="c28">YY</font>
 <span class="kw">and </span><br/><a NAME="E20:32_1_1_2_5"/>
<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>
 <span class="kw">and </span><br/><a NAME="E21:32_1_1_2_5"/>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>;<br/>
<a NAME="E22:32_1_1_2_5"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">ggx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c27">gg</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">ggy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c27">gg</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1_2_5"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="ref" href="relat_1.html#T11" title="RELAT_1:th.11">RELAT_1:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">x9</font> = <font color="Maroon" title="c21">x</font>, <font color="Maroon" title="c31">y9</font> = <font color="Maroon" title="c22">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c28">YY</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_2_5"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E5:32_1_1_2_5"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="txt" href="#E7:32_1_1_2_5"><span class="lab"><font color="Green" title="E53">A49</font></span></a>, <a class="txt" href="#E10:32_1_1_2_5"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E18:32_1_1_2_5"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E24:32_1_1_2_5"/><span class="lab"><font color="Green" title="E59">A55</font></span>: 
<font color="Maroon" title="c29">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32_1_1_2_5"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E25:32_1_1_2_5"/><span class="lab"><font color="Green" title="E60">A56</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E26:32_1_1_2_5"/><span class="kw">then </span>
<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="c15">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E3:32_1_1_2_5"><span class="lab"><font color="Green" title="E51">A47</font></span></a>, <a class="txt" href="#E5:32_1_1_2_5"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="txt" href="#E8:32_1_1_2_5"><span class="lab"><font color="Green" title="E54">A50</font></span></a>, <a class="txt" href="#E10:32_1_1_2_5"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E14:32_1_1_2_5"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="txt" href="#E18:32_1_1_2_5"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="ref" href="rlsub_1.html#T28" title="RLSUB_1:th.28">RLSUB_1:28</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E27:32_1_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c21">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c31">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T13" title="RLSUB_1:th.13">RLSUB_1:13</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c22">y</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="c30">x9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c31">y9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:32_1_1_2_5"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E24:32_1_1_2_5"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<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">x9</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">y9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#D2" target="_self" title="HAHNBAN:def.2">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">y9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:32_1_1_2_5"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E24:32_1_1_2_5"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:32_1_1_2_5"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E24:32_1_1_2_5"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E25:32_1_1_2"/>
<font color="Maroon" title="c19">f9</font> is  <a href="hahnban.html#V3" title="HAHNBAN:attr.3">homogeneous</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c15">Y</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="hahnban.html#D3" target="_self" title="HAHNBAN:def.3">HAHNBAN:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c22">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x</font></span>)</span></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c23">XX</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:32_1_1_2_6"/><span class="lab"><font color="Green" title="E50">A57</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">XX</font>
 <span class="kw">and </span><br/><a NAME="E3:32_1_1_2_6"/><span class="lab"><font color="Green" title="E51">A58</font></span>: 
<font color="Maroon" title="c23">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">gg</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E5:32_1_1_2_6"/><span class="lab"><font color="Green" title="E52">A59</font></span>: 
<font color="Maroon" title="c23">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">gg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_2_6"><span class="lab"><font color="Green" title="E51">A58</font></span></a></span>;<br/>
<a NAME="E6:32_1_1_2_6"/>
<font color="Maroon" title="c24">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">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="c26">ff</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c25">YY</font><span class="kw"> such that </span><br/><a NAME="E8:32_1_1_2_6"/><span class="lab"><font color="Green" title="E53">A60</font></span>: 
<font color="Maroon" title="c24">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">ff</font>
 <span class="kw">and </span><br/><a NAME="E9:32_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="c25">YY</font>
 <span class="kw">and </span><br/><a NAME="E10:32_1_1_2_6"/>
<font color="Maroon" title="c26">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>
 <span class="kw">and </span><br/><a NAME="E11:32_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="c25">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="c26">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">x9</font> = <font color="Maroon" title="c21">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c25">YY</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_2_6"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="txt" href="#E5:32_1_1_2_6"><span class="lab"><font color="Green" title="E52">A59</font></span></a>, <a class="txt" href="#E8:32_1_1_2_6"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E13:32_1_1_2_6"/><span class="lab"><font color="Green" title="E54">A61</font></span>: 
<font color="Maroon" title="c26">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1_2_6"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E14:32_1_1_2_6"/><span class="lab"><font color="Green" title="E55">A62</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">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="c25">YY</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:32_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c25">YY</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c15">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E3:32_1_1_2_6"><span class="lab"><font color="Green" title="E51">A58</font></span></a>, <a class="txt" href="#E5:32_1_1_2_6"><span class="lab"><font color="Green" title="E52">A59</font></span></a>, <a class="txt" href="#E8:32_1_1_2_6"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="ref" href="rlsub_1.html#T28" title="RLSUB_1:th.28">RLSUB_1:28</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E16:32_1_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c22">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c27">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T14" title="RLSUB_1:th.14">RLSUB_1:14</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c21">x</font></span>)</span> = 
<font color="Maroon" title="c26">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c27">x9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_1_1_2_6"><span class="lab"><font color="Green" title="E55">A62</font></span></a>, <a class="txt" href="#E13:32_1_1_2_6"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c22">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#D3" target="_self" title="HAHNBAN:def.3">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c22">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_1_1_2_6"><span class="lab"><font color="Green" title="E55">A62</font></span></a>, <a class="txt" href="#E13:32_1_1_2_6"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">f9</font> = <font color="Maroon" title="c19">f9</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c15">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:32_1_1_2"><span class="lab"><font color="Green" title="E49">A45</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E50">A63</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c15">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> holds <br/><font color="Maroon" title="c21">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">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="b1">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c23">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v</font> implies <font color="Maroon" title="c21">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">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="c23">v</font> )</span><br/><span class="kw">assume </span><a NAME="E1:32_1_1_2_7"/><span class="lab"><font color="Green" title="E51">A64</font></span>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">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="c23">v</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c24">XX</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:32_1_1_2_7"/><span class="lab"><font color="Green" title="E52">A65</font></span>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">XX</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_2_7"/><span class="lab"><font color="Green" title="E53">A66</font></span>: 
<font color="Maroon" title="c24">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#NK2" title="RELAT_1:NK.2">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="c11">E</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c25">gg</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">E</font><span class="kw"> such that </span><br/><a NAME="E6:32_1_1_2_7"/><span class="lab"><font color="Green" title="E54">A67</font></span>: 
<font color="Maroon" title="c24">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c25">gg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_2_7"><span class="lab"><font color="Green" title="E53">A66</font></span></a></span>;<br/><a NAME="E7:32_1_1_2_7"/>
<font color="Maroon" title="c25">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">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="c27">ff</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c26">YY</font><span class="kw"> such that </span><br/><a NAME="E9:32_1_1_2_7"/><span class="lab"><font color="Green" title="E55">A68</font></span>: 
<font color="Maroon" title="c25">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">ff</font>
 <span class="kw">and </span><br/><a NAME="E10:32_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="c26">YY</font>
 <span class="kw">and </span><br/><a NAME="E11:32_1_1_2_7"/>
<font color="Maroon" title="c27">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>
 <span class="kw">and </span><br/><a NAME="E12:32_1_1_2_7"/><span class="lab"><font color="Green" title="E56">A69</font></span>: 
 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="c26">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="c27">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">y9</font> = <font color="Maroon" title="c22">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c26">YY</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_2_7"><span class="lab"><font color="Green" title="E52">A65</font></span></a>, <a class="txt" href="#E6:32_1_1_2_7"><span class="lab"><font color="Green" title="E54">A67</font></span></a>, <a class="txt" href="#E9:32_1_1_2_7"><span class="lab"><font color="Green" title="E55">A68</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E14:32_1_1_2_7"/><span class="lab"><font color="Green" title="E57">A70</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c27">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="c26">YY</font> &amp; <font color="Maroon" title="c27">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">f9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1_2_7"><span class="lab"><font color="Green" title="E55">A68</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E15:32_1_1_2_7"/>
<font color="Maroon" title="c27">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">y9</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="c23">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_2_7"><span class="lab"><font color="Green" title="E51">A64</font></span></a>, <a class="txt" href="#E12:32_1_1_2_7"><span class="lab"><font color="Green" title="E56">A69</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:32_1_1_2_7"/>
<font color="Maroon" title="c21">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">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="c23">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_1_1_2_7"><span class="lab"><font color="Green" title="E57">A70</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E28:32_1_1_2"/>
<font color="Maroon" title="c20">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">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="c23">ff</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c22">YY</font><span class="kw"> such that </span><br/><a NAME="E30:32_1_1_2"/><span class="lab"><font color="Green" title="E51">A71</font></span>: 
<font color="Maroon" title="c20">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">ff</font>
 <span class="kw">and </span><br/><a NAME="E31:32_1_1_2"/><span class="lab"><font color="Green" title="E52">A72</font></span>: 
 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="c22">YY</font>
 <span class="kw">and </span><br/><a NAME="E32:32_1_1_2"/><span class="lab"><font color="Green" title="E53">A73</font></span>: 
<font color="Maroon" title="c23">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>
 <span class="kw">and </span><br/><a NAME="E33:32_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="c22">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="c23">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>;<br/><a NAME="E34:32_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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c23">ff</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:32_1_1_2"><span class="lab"><font color="Green" title="E52">A72</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E35:32_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A74</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:32_1_1_2"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="txt" href="#E32:32_1_1_2"><span class="lab"><font color="Green" title="E53">A73</font></span></a>, <a class="ref" href="grfunc_1.html#T27" title="GRFUNC_1:th.27">GRFUNC_1:27</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E36:32_1_1_2"/><span class="lab"><font color="Green" title="E55">A75</font></span>: 
<font color="Maroon" title="c17">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E18:32_1_1_2"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E37:32_1_1_2"/>
<font color="Maroon" title="c17">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="c22">YY</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:32_1_1_2"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E30:32_1_1_2"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E38:32_1_1_2"/><span class="kw">then </span>
 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="c15">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E31:32_1_1_2"><span class="lab"><font color="Green" title="E52">A72</font></span></a>, <a class="txt" href="#E36:32_1_1_2"><span class="lab"><font color="Green" title="E55">A75</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E39:32_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:32_1_1_2"><span class="lab"><font color="Green" title="E54">A74</font></span></a>, <a class="txt" href="#E27:32_1_1_2"><span class="lab"><font color="Green" title="E50">A63</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">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="c8">A</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c25">u</font> = <font color="Maroon" title="c24">u</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">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="c9">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c25">u</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c25">u</font>] )</span><br/><span class="kw">assume </span><a NAME="E41:32_1_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c25">u</font>]</span><br/><span class="kw">hence </span><a NAME="E42:32_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c25">u</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E13:32"/><span class="lab"><font color="Green" title="E39">A76</font></span>: 
 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="c8">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/>
<span class="kw">consider </span><font color="Maroon" title="c9">psi</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font><span class="kw"> such that </span><br/><a NAME="E15:32"/><span class="lab"><font color="Green" title="E40">A77</font></span>: 
 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="c8">A</font>  st <font color="Maroon" title="c9">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="c9">psi</font>,<font color="Olive" title="b1">chi</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="orders_1.html#S1" title="ORDERS_1:sch.1">ORDERS_1:sch 1</a>(<a class="txt" href="#E13:32"><span class="lab"><font color="Green" title="E39">A76</font></span></a>, <a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E11:32"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E38">A4</font></span></a>);<br/></span>
<a NAME="E16:32"/>
<font color="Maroon" title="c9">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)<span class="kw"> such that </span><br/><a NAME="E18:32"/><span class="lab"><font color="Green" title="E41">A78</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">psi</font>
 <span class="kw">and </span><br/><a NAME="E19:32"/><span class="lab"><font color="Green" title="E42">A79</font></span>: 
 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="c10">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">f9</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="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f9</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">f9</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/>
<span class="kw">consider </span><font color="Maroon" title="c11">Y</font> being    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E21:32"/><span class="lab"><font color="Green" title="E43">A80</font></span>: 
 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="c11">Y</font>
 <span class="kw">and </span><br/><a NAME="E22:32"/><span class="lab"><font color="Green" title="E44">A81</font></span>: 
<font color="Maroon" title="c10">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>
 <span class="kw">and </span><br/><a NAME="E23:32"/><span class="lab"><font color="Green" title="E45">A82</font></span>: 
 ex <font color="Olive" title="b1">f9</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c11">Y</font> st <br/>( <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f9</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="c11">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">f9</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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E42">A79</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">RLSY</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">Y</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c11">Y</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c11">Y</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c11">Y</font> #) as   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="hahnban.html#E30"><span class="lab"><font color="Green" title="E30">Lm2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">f9</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c11">Y</font><span class="kw"> such that </span><br/><a NAME="E26:32"/><span class="lab"><font color="Green" title="E46">A83</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f9</font>
 <span class="kw">and </span><br/><a NAME="E27:32"/><span class="lab"><font color="Green" title="E47">A84</font></span>: 
 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="c11">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">f9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:32"><span class="lab"><font color="Green" title="E45">A82</font></span></a></span>;<br/>
<a NAME="E28:32"/><span class="lab"><font color="Green" title="E48">A85</font></span>: 
<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="c6">RLS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="hahnban.html#E31"><span class="lab"><font color="Green" title="E31">Lm3</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E49">A86</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_2"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">Y</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c11">Y</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c11">Y</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c11">Y</font> #) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">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">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A87</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:32"><span class="lab"><font color="Green" title="E48">A85</font></span></a>, <a class="ref" href="rlsub_1.html#T32" title="RLSUB_1:th.32">RLSUB_1:32</a></span>;<br/><a NAME="E3:32_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></span>;<br/><a NAME="E4:32_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_2"><span class="lab"><font color="Green" title="E50">A87</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:32_2"/><span class="lab"><font color="Green" title="E51">A88</font></span>: 
<font color="Maroon" title="c14">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>
 <span class="kw">and </span><br/><a NAME="E7:32_2"/><span class="lab"><font color="Green" title="E52">A89</font></span>: 
 not <font color="Maroon" title="c14">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="c11">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_2"><span class="lab"><font color="Green" title="E51">A88</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">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="c11">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="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:32_2"/><span class="lab"><font color="Green" title="E53">A90</font></span>: 
<font color="Maroon" title="c16">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="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f9</font>
 <span class="kw">and </span><br/><a NAME="E11:32_2"/><span class="lab"><font color="Green" title="E54">A91</font></span>: 
 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="c11">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="c15">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="c16">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:32"><span class="lab"><font color="Green" title="E47">A84</font></span></a>, <a class="txt" href="#E7:32_2"><span class="lab"><font color="Green" title="E52">A89</font></span></a>, <a class="txt" href="hahnban.html#E29"><span class="lab"><font color="Green" title="E29">Lm1</font></span></a></span>;<br/><a NAME="E12:32_2"/><span class="lab"><font color="Green" title="E55">A92</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E13:32_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">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="c11">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="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#T1" target="_self" title="HAHNBAN:th.1">Th14</a></span>;<br/><a NAME="E14:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A93</font></span>: 
 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="c11">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="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32"><span class="lab"><font color="Green" title="E43">A80</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E15:32_2"/><span class="lab"><font color="Green" title="E57">A94</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">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="c11">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="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E16:32_2"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">phi</font> = <font color="Maroon" title="c16">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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_2"><span class="lab"><font color="Green" title="E55">A92</font></span></a>, <a class="ref" href="partfun1.html#D3" title="PARTFUN1:def.3">PARTFUN1:def 3</a></span>;<br/><a NAME="E18:32_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32"><span class="lab"><font color="Green" title="E43">A80</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E19:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:32"><span class="lab"><font color="Green" title="E44">A81</font></span></a>, <a class="txt" href="#E26:32"><span class="lab"><font color="Green" title="E46">A83</font></span></a>, <a class="txt" href="#E10:32_2"><span class="lab"><font color="Green" title="E53">A90</font></span></a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/><a NAME="E20:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A95</font></span>: 
<font color="Maroon" title="c17">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_2"><span class="lab"><font color="Green" title="E54">A91</font></span></a>, <a class="txt" href="#E14:32_2"><span class="lab"><font color="Green" title="E56">A93</font></span></a></span>;<br/><a NAME="E21:32_2"/>
<font color="Maroon" title="c15">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="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_4.html#T9" title="RLVECT_4:th.9">RLVECT_4:9</a></span>;<br/><a NAME="E22:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A96</font></span>: 
<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 <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="c15">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">phi</font> = <font color="Maroon" title="c17">phi</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32_2"><span class="lab"><font color="Green" title="E58">A95</font></span></a></span>;<br/><a NAME="E24:32_2"/>
 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="c15">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="c15">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="c11">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#T1" target="_self" title="HAHNBAN:th.1">Th14</a></span>;<br/><a NAME="E25:32_2"/><span class="kw">then </span>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">f9</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="c11">Y</font> &amp; <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 <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="c15">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="c11">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:32_2"><span class="lab"><font color="Green" title="E59">A96</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E26:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">phi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">psi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32"><span class="lab"><font color="Green" title="E41">A78</font></span></a>, <a class="txt" href="#E26:32"><span class="lab"><font color="Green" title="E46">A83</font></span></a>, <a class="txt" href="#E7:32_2"><span class="lab"><font color="Green" title="E52">A89</font></span></a>, <a class="txt" href="#E15:32_2"><span class="lab"><font color="Green" title="E57">A94</font></span></a>, <a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</a></span>;<br/><span class="kw">hence </span><a NAME="E27:32_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E40">A77</font></span></a>, <a class="txt" href="#E18:32"><span class="lab"><font color="Green" title="E41">A78</font></span></a>, <a class="txt" href="#E26:32"><span class="lab"><font color="Green" title="E46">A83</font></span></a>, <a class="txt" href="#E10:32_2"><span class="lab"><font color="Green" title="E53">A90</font></span></a>, <a class="ref" href="relat_1.html#T59" title="RELAT_1:th.59">RELAT_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c14">ggh</font> = <font color="Maroon" title="c13">f9</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c12">RLSY</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="hahnban.html#E33"><span class="lab"><font color="Green" title="E33">Lm5</font></span></a></span>;<br/>
<a NAME="E31:32"/>
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ggh</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:32"><span class="lab"><font color="Green" title="E46">A83</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">psi</font> = <font color="Maroon" title="c9">psi</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32"><span class="lab"><font color="Green" title="E41">A78</font></span></a>, <a class="txt" href="#E29:32"><span class="lab"><font color="Green" title="E49">A86</font></span></a>, <a class="txt" href="hahnban.html#E32"><span class="lab"><font color="Green" title="E32">Lm4</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">psi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  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="c1">V</font> holds  <font color="Maroon" title="c15">psi</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="b1">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E33:32"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32"><span class="lab"><font color="Green" title="E41">A78</font></span></a>, <a class="txt" href="#E22:32"><span class="lab"><font color="Green" title="E44">A81</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c15">psi</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="b1">x</font></span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">thus </span><a NAME="E34:32"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32"><span class="lab"><font color="Green" title="E41">A78</font></span></a>, <a class="txt" href="#E23:32"><span class="lab"><font color="Green" title="E45">A82</font></span></a>, <a class="txt" href="#E29:32"><span class="lab"><font color="Green" title="E49">A86</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
