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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>  st <font color="Olive" title="b3">f</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Olive" title="b3">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  &amp; <font color="Olive" title="b3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Olive" title="b3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">f</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
( <font color="Maroon" title="c4">f</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  )
 <span class="kw">and </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<a NAME="E3:20"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> implies <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:20_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">X1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E5:20_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c7">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="vectsp_5.html#D3" onmouseover="rs('vectsp_5/D3')" onmouseout="rh()">VECTSP_5:def 3</a></span>;<br/>
<a NAME="E6:20_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">BB</font> = <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>;<br/>
<a NAME="E1:20_1_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> &amp; <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c10">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> &amp; <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> implies <font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_1_1_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c9">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 <span class="kw">and </span><br/><a NAME="E2:20_1_1_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E4:20_1_1_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 <span class="kw">and </span><br/><a NAME="E5:20_1_1_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E6:20_1_1_1"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E8:20_1_1_1"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 <span class="kw">and </span><br/><a NAME="E9:20_1_1_1"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E10:20_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_1_1_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E11:20_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1_1_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="vectsp_4.html#T20" onmouseover="rs('vectsp_4/T20')" onmouseout="rh()">VECTSP_4:20</a></span>;<br/>
<a NAME="E12:20_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:20_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:20_1_1_1"/>
<font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E9:20_1_1_1"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E5:20_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="vectsp_1.html#D20" onmouseover="rs('vectsp_1/D20')" onmouseout="rh()">VECTSP_1:def 20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:20_1_1"/>
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> holds <br/><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> holds <br/><font color="Maroon" title="c9">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> implies <font color="Maroon" title="c9">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:20_1_1_2"/>
<font color="Maroon" title="c10">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E3:20_1_1_2"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 <span class="kw">and </span><br/><a NAME="E4:20_1_1_2"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E5:20_1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1_1_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E6:20_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T21" onmouseover="rs('vectsp_4/T21')" onmouseout="rh()">VECTSP_4:21</a></span>;<br/>
<a NAME="E7:20_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E8:20_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:20_1_1_2"/>
<font color="Maroon" title="c9">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:20_1_1_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="mod_2.html#D2" onmouseover="rs('mod_2/D2')" onmouseout="rh()">MOD_2:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:20_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="vectsp_4.html#D1" onmouseover="rs('vectsp_4/D1')" onmouseout="rh()">VECTSP_4:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">A1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">B</font><span class="kw"> such that </span><br/><a NAME="E8:20_1"/><span class="lab"><font color="Green" title="E19">A18</font></span>: 
<font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 ;<br/>
<a NAME="E9:20_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">A</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T17" onmouseover="rs('vectsp_4/T17')" onmouseout="rh()">VECTSP_4:17</a></span>;<br/>
<a NAME="E10:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A19</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E11:20_1"/><span class="lab"><font color="Green" title="E21">A20</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:20_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
;<br/>
<a NAME="E13:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</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="#E11:20_1"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E10:20_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E14:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A21</font></span>: 
 ex <font color="Olive" title="b1">B1</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">B</font> st  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="vectsp_4.html#T34" onmouseover="rs('vectsp_4/T34')" onmouseout="rh()">VECTSP_4:34</a></span>;<br/>
<a NAME="E15:20_1"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c8">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="ref" href="vectsp_8.html#D7" target="_self" onmouseover="rs('vectsp_8/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">X2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E17:20_1"/><span class="lab"><font color="Green" title="E24">A23</font></span>: 
<font color="Maroon" title="c9">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="vectsp_5.html#D3" onmouseover="rs('vectsp_5/D3')" onmouseout="rh()">VECTSP_5:def 3</a></span>;<br/>
<a NAME="E18:20_1"/><span class="lab"><font color="Green" title="E25">A24</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c10">BB</font> = <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>;<br/>
<a NAME="E1:20_1_2"/><span class="lab"><font color="Green" title="E26">A25</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> &amp; <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">v</font>, <font color="Maroon" title="c12">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> &amp; <font color="Maroon" title="c12">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> implies <font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_1_2_1"/><span class="lab"><font color="Green" title="E27">A26</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 <span class="kw">and </span><br/><a NAME="E2:20_1_2_1"/><span class="lab"><font color="Green" title="E28">A27</font></span>: 
<font color="Maroon" title="c12">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E4:20_1_2_1"/><span class="lab"><font color="Green" title="E29">A28</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 <span class="kw">and </span><br/><a NAME="E5:20_1_2_1"/><span class="lab"><font color="Green" title="E30">A29</font></span>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_2_1"><span class="lab"><font color="Green" title="E28">A27</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E6:20_1_2_1"/><span class="lab"><font color="Green" title="E31">A30</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_2_1"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E8:20_1_2_1"/><span class="lab"><font color="Green" title="E32">A31</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 <span class="kw">and </span><br/><a NAME="E9:20_1_2_1"/><span class="lab"><font color="Green" title="E33">A32</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_2_1"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E10:20_1_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_1_2_1"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E11:20_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1_2_1"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="ref" href="vectsp_4.html#T20" onmouseover="rs('vectsp_4/T20')" onmouseout="rh()">VECTSP_4:20</a></span>;<br/>
<a NAME="E12:20_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:20_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:20_1_2_1"/>
<font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E9:20_1_2_1"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="txt" href="#E5:20_1_2_1"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="ref" href="vectsp_1.html#D20" onmouseover="rs('vectsp_1/D20')" onmouseout="rh()">VECTSP_1:def 20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:20_1_2"/>
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> holds <br/><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> holds <br/><font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> implies <font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c12">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:20_1_2_2"/>
<font color="Maroon" title="c12">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c12">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E3:20_1_2_2"/><span class="lab"><font color="Green" title="E27">A33</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 <span class="kw">and </span><br/><a NAME="E4:20_1_2_2"/><span class="lab"><font color="Green" title="E28">A34</font></span>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E5:20_1_2_2"/>
<font color="Maroon" title="c13">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1_2_2"><span class="lab"><font color="Green" title="E27">A33</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E6:20_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c13">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T21" onmouseover="rs('vectsp_4/T21')" onmouseout="rh()">VECTSP_4:21</a></span>;<br/>
<a NAME="E7:20_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E8:20_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:20_1_2_2"/>
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c12">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:20_1_2_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="mod_2.html#D2" onmouseover="rs('mod_2/D2')" onmouseout="rh()">MOD_2:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:20_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_2"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="ref" href="vectsp_4.html#D1" onmouseover="rs('vectsp_4/D1')" onmouseout="rh()">VECTSP_4:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c10">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">B</font><span class="kw"> such that </span><br/><a NAME="E20:20_1"/><span class="lab"><font color="Green" title="E26">A35</font></span>: 
<font color="Maroon" title="c10">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 ;<br/>
<a NAME="E21:20_1"/><span class="lab"><font color="Green" title="E27">A36</font></span>: 
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c10">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20_1"><span class="lab"><font color="Green" title="E26">A35</font></span></a>, <a class="ref" href="vectsp_8.html#D7" target="_self" onmouseover="rs('vectsp_8/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E22:20_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">A</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T17" onmouseover="rs('vectsp_4/T17')" onmouseout="rh()">VECTSP_4:17</a></span>;<br/>
<a NAME="E23:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A37</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E24:20_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
;<br/>
<a NAME="E25:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</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="#E11:20_1"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E23:20_1"><span class="lab"><font color="Green" title="E28">A37</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">B2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">B</font><span class="kw"> such that </span><br/><a NAME="E27:20_1"/><span class="lab"><font color="Green" title="E29">A38</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:20_1"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="ref" href="vectsp_4.html#T34" onmouseover="rs('vectsp_4/T34')" onmouseout="rh()">VECTSP_4:34</a></span>;<br/>
<a NAME="E28:20_1"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:20_1"><span class="lab"><font color="Green" title="E29">A38</font></span></a>, <a class="ref" href="vectsp_7.html#T11" onmouseover="rs('vectsp_7/T11')" onmouseout="rh()">VECTSP_7:11</a></span>;<br/>
<a NAME="E29:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A39</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E17:20_1"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="txt" href="#E8:20_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E20:20_1"><span class="lab"><font color="Green" title="E26">A35</font></span></a>, <a class="txt" href="#E15:20_1"><span class="lab"><font color="Green" title="E23">A22</font></span></a>, <a class="txt" href="#E21:20_1"><span class="lab"><font color="Green" title="E27">A36</font></span></a>, <a class="txt" href="#E14:20_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E27:20_1"><span class="lab"><font color="Green" title="E29">A38</font></span></a>, <a class="ref" href="vectsp_7.html#T11" onmouseover="rs('vectsp_7/T11')" onmouseout="rh()">VECTSP_7:11</a></span>;<br/>
<a NAME="E30:20_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_1"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="ref" href="vectsp_4.html#D2" onmouseover="rs('vectsp_4/D2')" onmouseout="rh()">VECTSP_4:def 2</a></span>;<br/>
<a NAME="E31:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A40</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">X2</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="c7">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E29:20_1"><span class="lab"><font color="Green" title="E30">A39</font></span></a>, <a class="ref" href="funct_1.html#T87" onmouseover="rs('funct_1/T87')" onmouseout="rh()">FUNCT_1:87</a></span>;<br/>
<a NAME="E32:20_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_1"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="ref" href="vectsp_4.html#D2" onmouseover="rs('vectsp_4/D2')" onmouseout="rh()">VECTSP_4:def 2</a></span>;<br/>
<a NAME="E33:20_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</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="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E29:20_1"><span class="lab"><font color="Green" title="E30">A39</font></span></a>, <a class="ref" href="funct_1.html#T87" onmouseover="rs('funct_1/T87')" onmouseout="rh()">FUNCT_1:87</a></span>;<br/>
<a NAME="E34:20_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X1</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="c9">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:20_1"><span class="lab"><font color="Green" title="E31">A40</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:20_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E17:20_1"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="ref" href="vectsp_4.html#T29" onmouseover="rs('vectsp_4/T29')" onmouseout="rh()">VECTSP_4:29</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:20"/>
 <a href="vectsp_8.html#K6" title="VECTSP_8:func.6">FuncLatt</a> <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
