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<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="euclid_7.html#D8" target="_self" title="EUCLID_7:def.8">EUCLID_7:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font> )</span><br/>



<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E2:45_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<a NAME="E3:45_1_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4" title="SUBSET_1:def.4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E4:45_1_1"/><b>then </b>
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">Y</font> is <a href="euclid_7.html#V6" title="EUCLID_7:attr.6">linear_manifold</a> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">Z</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">Y</font> is <a href="euclid_7.html#V6" title="EUCLID_7:attr.6">linear_manifold</a> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) </span>}</span>  as  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>  ;<br/>
<a NAME="E6:45_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c7">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E7:45_1_1"/>
for <font color="Olive" title="b1">Y4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">Y</font> where <font color="Olive" title="b2">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b2">Y</font> is <a href="euclid_7.html#V6" title="EUCLID_7:attr.6">linear_manifold</a> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> ) </span>}</span>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">Y4</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">Y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">Y</font> is <a href="euclid_7.html#V6" title="EUCLID_7:attr.6">linear_manifold</a> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) </span>}</span>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y4</font> )</span><br/>

<b>assume </b><a NAME="E1:45_1_1_1"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c8">Y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">Y</font> is <a href="euclid_7.html#V6" title="EUCLID_7:attr.6">linear_manifold</a> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y4</font></span><br/>

<a NAME="E2:45_1_1_1"/><b>then </b><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y4</font>
 
<b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E3:45_1_1_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y4</font>
 
<b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E1:45_1_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:45_1_1_1"/>
 ex <font color="Olive" title="b1">Y0</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Maroon" title="c8">Y4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y0</font> &amp; <font color="Olive" title="b1">Y0</font> is <a href="euclid_7.html#V6" title="EUCLID_7:attr.6">linear_manifold</a> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y0</font> )
 
<b>by </b><i><a class="txt" href="#E1:45_1_1_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:45_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y4</font>
 <b>by </b><i><a class="txt" href="#E2:45_1_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E3:45_1_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="euclid_7.html#D8" target="_self" title="EUCLID_7:def.8">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:45_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_7.html#K3" title="EUCLID_7:func.3">L_Span</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="#E6:45_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
