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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="zmodul06.html#V3" title="ZMODUL06:attr.3">torsion-free</a>  <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>   <a href="zmodul03.html#V1" title="ZMODUL03:attr.1">finite-rank</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b3">I1</font> being   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Olive" title="b1">W1</font>  ex <font color="Olive" title="b4">I</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b4">I</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Olive" title="b2">W2</font></span>)</span> &amp; <font color="Olive" title="b3">I1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">I</font> &amp;  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Olive" title="b2">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b4">I</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>   <a href="zmodul03.html#V1" title="ZMODUL03:attr.1">finite-rank</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I1</font> being   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c2">W1</font>  ex <font color="Olive" title="b2">I</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">I</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> &amp; <font color="Olive" title="b1">I1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">I</font> &amp;  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b2">I</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">I1</font> be   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Maroon" title="c2">W1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">I</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> &amp; <font color="Maroon" title="c4">I1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> &amp;  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">I</font></span>)</span> )</span><br/>





<a NAME="E1:13"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c2">W1</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T97" onmouseover="rs('zmodul01/T97')" onmouseout="rh()">ZMODUL01:97</a></span>;<br/>
<a NAME="E2:13"/><span class="kw">then </span>
( <font color="Maroon" title="c4">I1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W1</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" onmouseover="rs('vectsp_4/D2')" onmouseout="rh()">VECTSP_4:def 2</a></span>;<br/>
<a NAME="E3:13"/><span class="kw">then </span>
<font color="Maroon" title="c4">I1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">II1</font> = <font color="Maroon" title="c4">I1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">II1</font> = <font color="Maroon" title="c5">II1</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">II1</font> = <font color="Maroon" title="c6">II1</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="vectsp_7.html#D3" onmouseover="rs('vectsp_7/D3')" onmouseout="rh()">VECTSP_7:def 3</a>, <a class="ref" href="zmodul03.html#T15" onmouseover="rs('zmodul03/T15')" onmouseout="rh()">ZMODUL03:15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">II</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:13"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
( <font color="Maroon" title="c7">II1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">II</font> &amp;  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">II</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T6" target="_self" onmouseover="rs('zmodul07/T6')" onmouseout="rh()">LmRankSX1</a></span>;<br/>
<a NAME="E9:13"/>
( <font color="Maroon" title="c8">II</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" onmouseover="rs('vectsp_4/D2')" onmouseout="rh()">VECTSP_4:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">I</font> = <font color="Maroon" title="c8">II</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">I</font> = <font color="Maroon" title="c9">I</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul03.html#T15" onmouseover="rs('zmodul03/T15')" onmouseout="rh()">ZMODUL03:15</a></span>;<br/>
<a NAME="E12:13"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c10">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="zmodul05.html#T3" onmouseover="rs('zmodul05/T3')" onmouseout="rh()">ZMODUL05:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:13"/>
 ex <font color="Olive" title="b1">I</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">I</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> &amp; <font color="Maroon" title="c4">I1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> &amp;  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">I</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E12">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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