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<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="matrlin.html#V2">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>







<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font> ;<br/>
<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W1</font>, <font color="Maroon">W2</font> = <font color="Maroon">W2</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E3:39"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font> is <a href="matrlin.html#V2">finite-dimensional</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">I</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span><b> such that </b><br/><a NAME="E5:39"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">I</font> is    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:39"/><i><font color="Green">E42</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E3:39"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E7:39"/>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T20">VECTSP_5:20</a></i>;<br/>
<b>then consider </b><font color="Maroon">I1</font> being    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">W1</font><b> such that </b><br/><a NAME="E9:39"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">I1</font> = <font color="Maroon">I1</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W1</font> <b>by </b><i/>;<br/>
<a NAME="E11:39"/>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T20">VECTSP_5:20</a></i>;<br/>
<b>then consider </b><font color="Maroon">I2</font> being    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">W2</font><b> such that </b><br/><a NAME="E13:39"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I2</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">I2</font> = <font color="Maroon">I2</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W2</font> <b>by </b><i/>;<br/>
<a NAME="E15:39"/><i><font color="Green">E45</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">I2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:39"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> is <a href="matrlin.html#V2">finite-dimensional</a>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">A</font> = <font color="Maroon">I1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">I2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">I2</font>
 ;<br/><a NAME="E2:39_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:39_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> )
 ;<br/><a NAME="E4:39_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E6:39_1"/>
( <font color="Maroon">v'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> or <font color="Maroon">v'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E7:39_1"/><b>then </b>
<font color="Maroon">v'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T6">VECTSP_5:6</a></i>;<br/><b>hence </b><a NAME="E8:39_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">I1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">I2</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E19:39"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/>
not <font color="Maroon">I1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I2</font>
 <b>and </b><br/><a NAME="E4:39_2"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:39_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:39_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I1</font> &amp; <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I2</font> )
 <b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/><a NAME="E7:39_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">W1</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">W1</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">W1</font> #) &amp; <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">W2</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">W2</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">W2</font> #) )
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><a NAME="E8:39_2"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E9:39_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:39_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D2">VECTSP_5:def 2</a></i>;<br/><a NAME="E11:39_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span>,the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E12:39_2"/><i><font color="Green">E74</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E6:39"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E14:39_2"/>
( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">I'</font> = <font color="Maroon">I</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:39_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:39_2_1"/>
( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><a NAME="E4:39_2_1"/><b>then </b>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:39_2_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E15:39"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:39_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E6:39"><i><font color="Green">E42</font></i></a>, <a class="txt" href="vectsp_9.html#E13:39"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Ix</font> = <font color="Maroon">I</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E18:39_2"/>
( <font color="Maroon">I1</font> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp; <font color="Maroon">I1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W1</font> )
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">I1'</font> = <font color="Maroon">I1</font> as  <a href="vectsp_7.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:39_2_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:39_2_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:39_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1'</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E21:39_2"/><b>then </b>
<font color="Maroon">Ix</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E22:39_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">Ix</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T2">VECTSP_7:2</a></i>;<br/><a NAME="E23:39_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E24:39_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ix</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><font color="Maroon">Ix</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> = 
<font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>
;<br/><a NAME="E26:39_2"/><b>then </b><i><font color="Green">E78</font></i>: 
not <font color="Maroon">x'</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I'</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E15:39"><i><font color="Green">E45</font></i></a>, <a class="txt" href="vectsp_9.html#E16:39"><i><font color="Green">E46</font></i></a>, </i>;<br/><a NAME="E27:39_2"/>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E28:39_2"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, <a class="txt" href="vectsp_9.html#E9:39"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:39"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E22:39"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T64">CARD_2:64</a></i>;<br/>
<a NAME="E23:39"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E25:39"/><i><font color="Green">E80</font></i>: 
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span>
 ;<br/><a NAME="E2:39_3"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">w1</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">w2</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:39_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">w1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E5:39_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">w2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 <b>and </b><br/><a NAME="E6:39_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/><b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w1</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W1</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E6:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">w2</font> = <font color="Maroon">w2</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W2</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E9:39"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E9:39_3"/>
<font color="Maroon">w1</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I1</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">K1</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">I1</font><b> such that </b><br/><a NAME="E11:39_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K1</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/><b>consider </b><font color="Maroon">L1</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E13:39_3"/><i><font color="Green">E86</font></i>: 
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K1</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K1</font> )
 <b>by </b><i/>;<br/><a NAME="E14:39_3"/>
<font color="Maroon">w2</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I2</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">K2</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">I2</font><b> such that </b><br/><a NAME="E16:39_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K2</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/><b>consider </b><font color="Maroon">L2</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E18:39_3"/><i><font color="Green">E88</font></i>: 
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K2</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K2</font> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">L</font> = <font color="Maroon">L1</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font>;<br/><a NAME="E19:39_3"/>
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1</font> &amp;  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I2</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E16:39"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><a NAME="E20:39_3"/><b>then </b>
(  <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">I2</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T51">VECTSP_6:51</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/><a NAME="E21:39_3"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">I2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L1</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A'</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><a NAME="E23:39_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E13:39"><i><font color="Green">E44</font></i></a>, <a class="txt" href="vectsp_9.html#E15:39"><i><font color="Green">E45</font></i></a>, <a class="txt" href="vectsp_9.html#E16:39"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="vectsp_6.html#T77">VECTSP_6:77</a></i>;<br/><a NAME="E24:39_3"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A'</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/><b>hence </b><a NAME="E25:39_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A'</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:39"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E28:39"/><b>then </b>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T35">VECTSP_4:35</a></i>;<br/>
<a NAME="E29:39"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T33">VECTSP_4:33</a></i>;<br/>
<a NAME="E30:39"/>
for <font color="Olive">L</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font>  st  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> holds <br/> <a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">L</font> be    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:39_4"/><i><font color="Green">E92</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span>
 ;<br/>

<a NAME="E2:39_4"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W2</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T11">VECTSP_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W1'</font> = <font color="Maroon">W1</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> <b>by </b><i><a class="ref" href="vectsp_5.html#T11">VECTSP_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W2'</font> = <font color="Maroon">W2</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> <b>by </b><i><a class="ref" href="vectsp_5.html#T11">VECTSP_5:11</a></i>;<br/>
<a NAME="E5:39_4"/><i><font color="Green">E95</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">I2</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span><b> such that </b><br/><a NAME="E7:39_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E8:39_4"/><i><font color="Green">E97</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E9:39_4"/><i><font color="Green">E136</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9">VECTSP_6:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I1</font>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I1</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E12:39_4"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E16:39"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_3.html#T124">FINSEQ_3:124</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span>, <font color="Maroon">F2</font> = <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <font color="Maroon">B</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_3.html#T93">FINSEQ_3:93</a></i>;<br/>
<a NAME="E14:39_4"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">F1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">F2</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E16:39"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_3.html#T94">FINSEQ_3:94</a></i>;<br/>
<b>consider </b><font color="Maroon">L1</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font><b> such that </b><br/><a NAME="E16:39_4"/><i><font color="Green">E139</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span>
 <b>and </b><br/><a NAME="E17:39_4"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">L1</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F1</font>
 <b>by </b><i/>;<br/>
<a NAME="E18:39_4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E19:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E19:39_4"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L1</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E20:39_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">X</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E23:39_4"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/>
<b>then </b><i><font color="Green">E142</font></i>:  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> = 
<font color="Maroon">I1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E19:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I1</font>

;<br/>
<a NAME="E25:39_4"/><b>then </b><i><font color="Green">E143</font></i>: 
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1</font> &amp; <font color="Maroon">I1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E26:39_4"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">K1</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">W1'</font><b> such that </b><br/><a NAME="E28:39_4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font>
 <b>and </b><br/><a NAME="E29:39_4"/><i><font color="Green">E166</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">L2</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font><b> such that </b><br/><a NAME="E31:39_4"/><i><font color="Green">E167</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span>
 <b>and </b><br/><a NAME="E32:39_4"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">L2</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F2</font>
 <b>by </b><i/>;<br/>
<a NAME="E33:39_4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E34:39_4"/><b>then </b><i><font color="Green">E169</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L2</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E35:39_4"/><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">I1</font> <a href="tarski.html#R1">c=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">I1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>
;<br/>
<a NAME="E37:39_4"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">I1</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E38:39_4"/><b>then </b>
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I2</font> &amp; <font color="Maroon">I2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> )
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E15:39"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/>
<a NAME="E39:39_4"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">K2</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">W2'</font><b> such that </b><br/><a NAME="E41:39_4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font>
 <b>and </b><br/><a NAME="E42:39_4"/><i><font color="Green">E172</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L2</font>
 <b>by </b><i/>;<br/>
<i><font color="Green">E173</font></i>:  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E9:39"><i><font color="Green">E43</font></i></a>, , , </i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T37">VECTSP_6:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
;<br/>
<b>then </b> <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L1</font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T63">VECTSP_1:63</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T23">VECTSP_4:23</a></i>
;<br/>
<a NAME="E45:39_4"/><b>then </b>
(  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E46:39_4"/><b>then </b>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7">VECTSP_5:7</a></i>;<br/>
<a NAME="E47:39_4"/><b>then </b>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K1</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, <a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">KI</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E49:39_4"/><i><font color="Green">E175</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KI</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/>
<a NAME="E50:39_4"/>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T29">VECTSP_5:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">LI</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font><b> such that </b><br/><a NAME="E52:39_4"/><i><font color="Green">E177</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LI</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KI</font>
 <b>and </b><br/><a NAME="E53:39_4"/><i><font color="Green">E178</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LI</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KI</font>
 <b>by </b><i/>;<br/>
<a NAME="E54:39_4"/><i><font color="Green">E179</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">LI</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LI</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T51">VECTSP_6:51</a></i>;<br/>
<a NAME="E55:39_4"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">I</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">I2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E56:39_4"/>
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LI</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> &amp;  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I2</font> )
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E15:39"><i><font color="Green">E45</font></i></a>, , , <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a>, <a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/>
<a NAME="E57:39_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LI</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E58:39_4"/><b>then </b><i><font color="Green">E181</font></i>: 
(  <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">LI</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I2</font> &amp; <font color="Maroon">I2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E59:39_4"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">LI</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">K</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">W2</font><b> such that </b><br/><a NAME="E61:39_4"/><i><font color="Green">E182</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">LI</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span>
 <b>and </b><br/><a NAME="E62:39_4"/><i><font color="Green">E183</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">LI</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E13:39"><i><font color="Green">E44</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">K</font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">I2</font> <b>by </b><i>, , <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E64:39_4"/>
( <font color="Maroon">I1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W1</font> &amp; <font color="Maroon">I1</font> is <a href="vectsp_7.html#V1">linearly-independent</a> )
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">I1'</font> = <font color="Maroon">I1</font> as  <a href="vectsp_7.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> <b>by </b><i><a class="txt" href="vectsp_9.html#E13:39"><i><font color="Green">E44</font></i></a>, </i>;<br/>
<a NAME="E66:39_4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LI</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E67:39_4"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LI</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I1'</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E68:39_4"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">LI</font> <a href="vectsp_6.html#R1">=</a> <font color="Maroon">L1</font>
 
<b>by </b><i>, , , <a class="ref" href="vectsp_9.html#T9" target="_self">Th9</a>, <a class="ref" href="matrlin.html#T9">MATRLIN:9</a></i>;<br/>
<a NAME="E69:39_4"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">I2</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W2</font> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LI</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="vectsp_9.html#T9" target="_self">Th9</a>, <a class="ref" href="vectsp_4.html#T20">VECTSP_4:20</a></i>
<br/>.= 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="vectsp_9.html#T10" target="_self">Th10</a>, <a class="ref" href="vectsp_6.html#T77">VECTSP_6:77</a></i>
;<br/>
<a NAME="E71:39_4"/><b>then </b><i><font color="Green">E186</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#T11" target="_self">Th11</a>, , <a class="ref" href="vectsp_7.html#D1">VECTSP_7:def 1</a></i>;<br/>
<a NAME="E72:39_4"/><i><font color="Green">E187</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T7" target="_self">Th7</a>, , <a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E73:39_4"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">I1</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">I1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">I1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">I1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_9.html#T7" target="_self">Th7</a>, , <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K8">/\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<a NAME="E75:39_4"/><b>then </b><i><font color="Green">E189</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="xboole_0.html#R1">misses</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_4_8"/>
not  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_4_8"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:39_4_8"/><i><font color="Green">E191</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span> <b>by </b><i/>;<br/><i><font color="Green">E192</font></i>:  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> = 
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="vectsp_9.html#E22:39"><i><font color="Green">E79</font></i></a>, <a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D11">VECTSP_6:def 11</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_4_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_4_8_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_4_8_2_1"><b>suppose </b></a><a NAME="E1:39_4_8_2_1"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span><b> such that </b><br/><a NAME="E3:39_4_8_2_1"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T19">VECTSP_6:19</a></i>;<br/><a NAME="E4:39_4_8_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:39_4_8_2_1"/><b>then </b>
<font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/><b>hence </b><a NAME="E6:39_4_8_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_9.html#E25:39"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_4_8_2_2"><b>suppose </b></a><a NAME="E1:39_4_8_2_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span><b> such that </b><br/><a NAME="E3:39_4_8_2_2"/><i><font color="Green">E196</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T19">VECTSP_6:19</a></i>;<br/><a NAME="E4:39_4_8_2_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:39_4_8_2_2"/><b>then </b>
<font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/><b>hence </b><a NAME="E6:39_4_8_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_9.html#E25:39"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E77:39_4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:39"/><b>then </b>
<font color="Maroon">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D1">VECTSP_7:def 1</a></i>;<br/>
<a NAME="E32:39"/><b>then </b>
<font color="Maroon">A</font> is    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E6:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green">E40</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W2</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E34:39"/>
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W2</font></span>)</span>
 ;<br/>


</div>
