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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">V1</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b2">V1</font>,<font color="Olive" title="b2">V1</font><br/> for <font color="Olive" title="b4">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> ex <font color="Olive" title="b5">h</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b2">V1</font>,<font color="Olive" title="b2">V1</font> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Olive" title="b2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b5">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Olive" title="b2">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; ( for <font color="Olive" title="b6">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b5">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b6">i</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V1</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b1">V1</font><br/> for <font color="Olive" title="b3">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> ex <font color="Olive" title="b4">h</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b1">V1</font> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Olive" title="b1">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b4">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Olive" title="b1">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b4">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b5">i</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">V1</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font><br/> for <font color="Olive" title="b2">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> ex <font color="Olive" title="b3">h</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b3">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b3">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b4">i</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> ex <font color="Olive" title="b2">h</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b2">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b2">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b3">i</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">L</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  ex <font color="Olive" title="b1">h</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b1">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> ) ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">h</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> $1 <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> $1</span>)</span> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b1">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> ) );<br/>
<a NAME="E1:56"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E2:56_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">h</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font><span class="kw"> such that </span><br/><a NAME="E4:56_1"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:56_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/>
<span class="kw">take </span><font color="Maroon" title="c9">H</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <font color="Maroon" title="c8">h</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c9">H</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">H</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> ) )</span><br/>

<a NAME="E6:56_1"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E48"><span class="lab"><font color="Green" title="E37">Lm6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E49"><span class="lab"><font color="Green" title="E38">Lm7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p0">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E50"><span class="lab"><font color="Green" title="E39">Lm8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p0">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E46"><span class="lab"><font color="Green" title="E35">Lm4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p0">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p0">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E47"><span class="lab"><font color="Green" title="E36">Lm5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E49"><span class="lab"><font color="Green" title="E38">Lm7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c9">H</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">H</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c9">H</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">H</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span><br/>

<span class="lab"><font color="Green" title="E47">A5</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></span>
;<br/>
<span class="lab"><font color="Green" title="E48">A6</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E51"><span class="lab"><font color="Green" title="E40">Lm9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E47"><span class="lab"><font color="Green" title="E36">Lm5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T18" target="_self" title="VECTSP11:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T18" target="_self" title="VECTSP11:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <font color="Maroon" title="c5">L</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E46"><span class="lab"><font color="Green" title="E35">Lm4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E51"><span class="lab"><font color="Green" title="E40">Lm9</font></span></a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <font color="Maroon" title="c8">h</font></span>)</span> = 
<font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E47"><span class="lab"><font color="Green" title="E36">Lm5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E46"><span class="lab"><font color="Green" title="E35">Lm4</font></span></a></span>
;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c9">H</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E49"><span class="lab"><font color="Green" title="E38">Lm7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E9:56_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="vectsp11.html#E49"><span class="lab"><font color="Green" title="E38">Lm7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p5">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E48"><span class="lab"><font color="Green" title="E37">Lm6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">H</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E48"><span class="lab"><font color="Green" title="E37">Lm6</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:56"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c7">h</font> =  <a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1 <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c7">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> ) )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1 = 
<font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T18" target="_self" title="VECTSP11:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c7">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c7">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c8">i</font></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c7">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T18" target="_self" title="VECTSP11:th.18">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c8">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></span>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c8">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T18" target="_self" title="VECTSP11:th.18">Th18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:56"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E43">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:56"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  ex <font color="Olive" title="b1">h</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="matrlin2.html#K9" title="MATRLIN2:func.9">*</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c3">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b1">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Olive" title="b2">i</font></span>)</span> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
