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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">K</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b2">V1</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>   <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>   <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>   <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">ModuleStr</a> over <font color="Olive" title="b1">K</font><br/>  for <font color="Olive" title="b3">W</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">V1</font><br/>  for <font color="Olive" title="b4">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">V1</font>,<font color="Olive" title="b2">V1</font><br/>  for <font color="Olive" title="b5">fW</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b3">W</font>,<font color="Olive" title="b3">W</font>  st <font color="Olive" title="b5">fW</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Olive" title="b3">W</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Olive" title="b3">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b5">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">V1</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>   <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>   <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>   <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">ModuleStr</a> over <font color="Maroon" title="c2">K</font><br/>  for <font color="Olive" title="b2">W</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V1</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b1">V1</font><br/>  for <font color="Olive" title="b4">fW</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">W</font>,<font color="Olive" title="b2">W</font>  st <font color="Olive" title="b4">fW</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Olive" title="b2">W</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Olive" title="b2">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V1</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>   <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>   <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>   <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">ModuleStr</a> over <font color="Maroon" title="c2">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">V1</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font><br/>  for <font color="Olive" title="b3">fW</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">W</font>,<font color="Olive" title="b1">W</font>  st <font color="Olive" title="b3">fW</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Olive" title="b1">W</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Olive" title="b1">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font><br/>  for <font color="Olive" title="b2">fW</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">W</font>,<font color="Maroon" title="c4">W</font>  st <font color="Olive" title="b2">fW</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">fW</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">W</font>,<font color="Maroon" title="c4">W</font>  st <font color="Olive" title="b1">fW</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">fW</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">W</font>,<font color="Maroon" title="c4">W</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"> ( <font color="Maroon" title="c6">fW</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font> )</span><br/>











<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c6">fW</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> $1</span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> $1;<br/>
<a NAME="E2:30"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( <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="E1:30_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</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"> <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/>

<a NAME="E2:30_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c7">n</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="c4">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</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> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" onmouseover="rs('vectsp11/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T83" onmouseover="rs('relat_1/T83')" onmouseout="rh()">RELAT_1:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" onmouseover="rs('vectsp11/T19')" onmouseout="rh()">Th19</a></span>
<br/>.= 
<font color="Maroon" title="c6">fW</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="relat_1.html#T65" onmouseover="rs('relat_1/T65')" onmouseout="rh()">RELAT_1:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" onmouseover="rs('vectsp11/T19')" onmouseout="rh()">Th19</a></span>
<br/>.= 
<font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</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="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" onmouseover="rs('vectsp11/T20')" onmouseout="rh()">Th20</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/>


</div><span class="kw">end;</span></div>
<a NAME="E3:30"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">W</font> <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="vectsp_4.html#D2" onmouseover="rs('vectsp_4/D2')" onmouseout="rh()">VECTSP_4:def 2</a></span>;<br/>
<a NAME="E4:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A5</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">W</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c3">V1</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T18" target="_self" onmouseover="rs('vectsp11/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c3">V1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c4">W</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T65" onmouseover="rs('relat_1/T65')" onmouseout="rh()">RELAT_1:65</a></span>
<br/>.= 
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c4">W</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="funct_1.html#T22" onmouseover="rs('funct_1/T22')" onmouseout="rh()">FUNCT_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T18" target="_self" onmouseover="rs('vectsp11/T18')" onmouseout="rh()">Th18</a></span>
;<br/>
<a NAME="E6:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E7:30"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E22">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:30"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrlin2.html#K7" title="MATRLIN2:func.7">|</a> <font color="Maroon" title="c4">W</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">fW</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c1">n</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"> verum</span><br/>


</div>
