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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</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">V</font>, <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">V1</font>, <font color="Olive" title="b4">W1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">V1</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W1</font> holds <br/><font color="Olive" title="b1">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b4">W1</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <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">V1</font>, <font color="Olive" title="b2">W1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V1</font> &amp; <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W1</font> holds <br/><font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Olive" title="b2">W1</font></span><br/><b>let </b><font color="Maroon" title="c4">V1</font>, <font color="Maroon" title="c5">W1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V1</font> &amp; <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> implies <font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font> )</span><br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V1</font> &amp; <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font></span><br/>

<b>thus </b><a NAME="E2:8"/>
<font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font> )</span><br/>

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v</font> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">u1</font> = <font color="Maroon" title="c7">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E3:8_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c8">v</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E3:8_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<a NAME="E6:8_1"/>
<font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">v1</font>
 
<b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:8_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E3:8_1"><i><font color="Green" title="E8">A4</font></i></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>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font> )</span><br/>

<b>assume </b><a NAME="E3:8"/><i><font color="Green" title="E7">B3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c5">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E5:8"/><i><font color="Green" title="E8">B4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v</font> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V1</font> &amp; <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> )
 <b>by </b><i><a class="txt" href="#E3:8"><i><font color="Green" title="E7">B3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">u1</font> = <font color="Maroon" title="c7">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E5:8"><i><font color="Green" title="E8">B4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c8">v</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E5:8"><i><font color="Green" title="E8">B4</font></i></a></i>;<br/>
<a NAME="E8:8"/>
<font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">v1</font>
 
<b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:8"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E5:8"><i><font color="Green" title="E8">B4</font></i></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>
