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<span class="kw">let </span><font color="Maroon" title="c1">V</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#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="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c1">V</font> holds   <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="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b2">W2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Olive" title="b1">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Olive" title="b2">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</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="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">  <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="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c3">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font></span>)</span></span><br/>



<a NAME="E1:16"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
(  <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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> &amp; <span class="p1">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font></span>)</span></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"><font color="Maroon" title="c2">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T2" onmouseover="rs('zmodul05/T2')" onmouseout="rh()">ZMODUL05:2</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c3">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p3">(<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>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul06.html#T62" onmouseover="rs('zmodul06/T62')" onmouseout="rh()">ZMODUL06:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p2">(<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>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E3:16"/>
 <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="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c2">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c3">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</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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