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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</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">V</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b4">U1</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Olive" title="b3">W2</font><br/>  for <font color="Olive" title="b5">U2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b2">W1</font>  st <font color="Olive" title="b4">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W2</font> &amp; <font color="Olive" title="b5">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b3">W2</font> holds <br/> ex <font color="Olive" title="b6">F</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Olive" title="b3">W2</font></span>)</span>,<font color="Olive" title="b4">U1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b2">W1</font>,<font color="Olive" title="b5">U2</font>)</span>)</span> st <font color="Olive" title="b6">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">R</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">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">U1</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b1">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Olive" title="b2">W2</font><br/>  for <font color="Olive" title="b4">U2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b1">W1</font>  st <font color="Olive" title="b3">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> &amp; <font color="Olive" title="b4">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b2">W2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Olive" title="b2">W2</font></span>)</span>,<font color="Olive" title="b3">U1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b1">W1</font>,<font color="Olive" title="b4">U2</font>)</span>)</span> st <font color="Olive" title="b5">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U1</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font><br/>  for <font color="Olive" title="b2">U2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c3">W1</font>  st <font color="Olive" title="b1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Olive" title="b2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> holds <br/> ex <font color="Olive" title="b3">F</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Olive" title="b1">U1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W1</font>,<font color="Olive" title="b2">U2</font>)</span>)</span> st <font color="Olive" title="b3">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">U1</font> be   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</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">U2</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c3">W1</font>  st <font color="Maroon" title="c5">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Olive" title="b1">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> holds <br/> ex <font color="Olive" title="b2">F</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W1</font>,<font color="Olive" title="b1">U2</font>)</span>)</span> st <font color="Olive" title="b2">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">U2</font> be   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c3">W1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c6">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> implies  ex <font color="Olive" title="b1">F</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c6">U2</font>)</span>)</span> st <font color="Olive" title="b1">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/>









<span class="kw">assume </span><a NAME="E1:64"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
( <font color="Maroon" title="c5">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c6">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c6">U2</font>)</span>)</span> st <font color="Olive" title="b1">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">Z1</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Z2</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c6">U2</font>);<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> );<br/>
<a NAME="E2:64"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font>  ex <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">z</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">zv</font> = <font color="Maroon" title="c9">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T25" onmouseover="rs('zmodul01/T25')" onmouseout="rh()">ZMODUL01:25</a></span>;<br/>
<a NAME="E2:64_1"/>
<font color="Maroon" title="c10">zv</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 
;<br/>
<a NAME="E3:64_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">zv</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T93" onmouseover="rs('zmodul01/T93')" onmouseout="rh()">ZMODUL01:93</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">zz</font> = <font color="Maroon" title="c10">zv</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c11">zz</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font>;<br/>
<a NAME="E5:64_1"/>
<font color="Maroon" title="c11">zz</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c5">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</a></span>;<br/>
<a NAME="E6:64_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">zz</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">yy</font> = <font color="Maroon" title="c11">zz</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">yy</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="c9">z</font>,<font color="Maroon" title="c13">yy</font>]</span><br/>

<span class="kw">thus </span><a NAME="E8:64_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">z</font>,<font color="Maroon" title="c13">yy</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><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:64"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E37">A2</font></span></a>);<br/></span>
<a NAME="E5:64"/>
<font color="Maroon" title="c9">f</font> is   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">W1</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:64_2"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font> holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">xx</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:64_2_1"/><span class="lab"><font color="Green" title="E39">C1</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">xx</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">xx</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">yy</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:64_2_1"/><span class="lab"><font color="Green" title="E40">C2</font></span>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">yy</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">yy</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">xy</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:64_2_1"/><span class="lab"><font color="Green" title="E41">C3</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xy</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xy</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">xv</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c16">yv</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T25" onmouseover="rs('zmodul01/T25')" onmouseout="rh()">ZMODUL01:25</a></span>;<br/>
<a NAME="E8:64_2_1"/>
( <font color="Maroon" title="c12">xx</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c5">U1</font> &amp; <font color="Maroon" title="c13">yy</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c5">U1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</a></span>;<br/>
<a NAME="E9:64_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c12">xx</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>) &amp; <font color="Maroon" title="c13">yy</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>) )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">xU</font> = <font color="Maroon" title="c12">xx</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font>, <font color="Maroon" title="c18">yU</font> = <font color="Maroon" title="c13">yy</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>) ;<br/>
<span class="lab"><font color="Green" title="E42">C4</font></span>: <font color="Maroon" title="c12">xx</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">yy</font> = 
<font color="Maroon" title="c15">xv</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">yv</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_1"><span class="lab"><font color="Green" title="E39">C1</font></span></a>, <a class="txt" href="#E4:64_2_1"><span class="lab"><font color="Green" title="E40">C2</font></span></a>, <a class="ref" href="zmodul01.html#T28" onmouseover="rs('zmodul01/T28')" onmouseout="rh()">ZMODUL01:28</a></span>
<br/>.= 
<font color="Maroon" title="c14">xy</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_2_1"><span class="lab"><font color="Green" title="E41">C3</font></span></a>, <a class="ref" href="zmodul01.html#T28" onmouseover="rs('zmodul01/T28')" onmouseout="rh()">ZMODUL01:28</a></span>
;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c17">xU</font>,<font color="Maroon" title="c18">yU</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_1"><span class="lab"><font color="Green" title="E39">C1</font></span></a>, <a class="txt" href="#E4:64_2_1"><span class="lab"><font color="Green" title="E40">C2</font></span></a>, <a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_2_1"><span class="lab"><font color="Green" title="E41">C3</font></span></a>, <a class="txt" href="#E11:64_2_1"><span class="lab"><font color="Green" title="E42">C4</font></span></a>, <a class="ref" href="vectsp10.html#D3" onmouseover="rs('vectsp10/D3')" onmouseout="rh()">VECTSP10:def 3</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="E2:64_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">B1</font></span>: 
<font color="Maroon" title="c9">f</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a> 
 
;<br/>
<a NAME="E3:64_2"/>
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font> holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font> holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c12">xx</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:64_2_2"/><span class="lab"><font color="Green" title="E40">C1</font></span>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">xx</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">xx</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">ax</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:64_2_2"/><span class="lab"><font color="Green" title="E41">C2</font></span>: 
( <font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ax</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ax</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">xv</font> = <font color="Maroon" title="c11">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T25" onmouseover="rs('zmodul01/T25')" onmouseout="rh()">ZMODUL01:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E42">C3</font></span>: <font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c12">xx</font> = 
<font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c14">xv</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_2"><span class="lab"><font color="Green" title="E40">C1</font></span></a>, <a class="ref" href="zmodul01.html#T29" onmouseover="rs('zmodul01/T29')" onmouseout="rh()">ZMODUL01:29</a></span>
<br/>.= 
<font color="Maroon" title="c13">ax</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_2_2"><span class="lab"><font color="Green" title="E41">C2</font></span></a>, <a class="ref" href="zmodul01.html#T29" onmouseover="rs('zmodul01/T29')" onmouseout="rh()">ZMODUL01:29</a></span>
;<br/>
<a NAME="E7:64_2_2"/>
<font color="Maroon" title="c12">xx</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c5">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</a></span>;<br/>
<a NAME="E8:64_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">xx</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">xU</font> = <font color="Maroon" title="c12">xx</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>) ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c15">xU</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_2_2"><span class="lab"><font color="Green" title="E41">C2</font></span></a>, <a class="txt" href="#E6:64_2_2"><span class="lab"><font color="Green" title="E42">C3</font></span></a>, <a class="ref" href="vectsp10.html#D5" onmouseover="rs('vectsp10/D5')" onmouseout="rh()">VECTSP10:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c10">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_2"><span class="lab"><font color="Green" title="E40">C1</font></span></a>, <a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:64_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E5:64_2"/>
<font color="Maroon" title="c9">f</font> is   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">W1</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2"><span class="lab"><font color="Green" title="E39">B1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">W1</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> ;<br/>
<a NAME="E7:64"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">U2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:64_3"/>
 for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font> holds <br/> ( <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f</font> iff <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">U2</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f</font> iff <font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">U2</font> )</span><br/>

<div><span class="kw">hereby </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"> ( <font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">U2</font> implies <font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:64_3_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f</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="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">U2</font></span><br/><a NAME="E2:64_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">C1</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T10" onmouseover="rs('ranknull/T10')" onmouseout="rh()">RANKNULL:10</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">vv</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:64_3_1_1"/><span class="lab"><font color="Green" title="E41">C2</font></span>: 
( <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vv</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vv</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/><font color="Maroon" title="c12">vv</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> = 
 <a href="vectsp10.html#K4" title="VECTSP10:func.4">zeroCoset</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_3_1_1"><span class="lab"><font color="Green" title="E40">C1</font></span></a>, <a class="txt" href="#E4:64_3_1_1"><span class="lab"><font color="Green" title="E41">C2</font></span></a>, <a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font>

;<br/><a NAME="E6:64_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">C3</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E4:64_3_1_1"><span class="lab"><font color="Green" title="E41">C2</font></span></a>, <a class="ref" href="zmodul01.html#T63" onmouseover="rs('zmodul01/T63')" onmouseout="rh()">ZMODUL01:63</a></span>;<br/><a NAME="E7:64_3_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 ;<br/><span class="kw">hence </span><a NAME="E8:64_3_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:64_3_1_1"><span class="lab"><font color="Green" title="E42">C3</font></span></a>, <a class="ref" href="zmodul01.html#T94" onmouseover="rs('zmodul01/T94')" onmouseout="rh()">ZMODUL01:94</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>

<span class="kw">assume </span><a NAME="E2:64_3_1"/><span class="lab"><font color="Green" title="E40">C1</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">U2</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="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">vv</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:64_3_1"/><span class="lab"><font color="Green" title="E41">C2</font></span>: 
( <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vv</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vv</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<a NAME="E5:64_3_1"/>
<font color="Maroon" title="c12">vv</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:64_3_1"><span class="lab"><font color="Green" title="E40">C1</font></span></a>, <a class="txt" href="#E4:64_3_1"><span class="lab"><font color="Green" title="E41">C2</font></span></a>, <a class="ref" href="zmodul01.html#T94" onmouseover="rs('zmodul01/T94')" onmouseout="rh()">ZMODUL01:94</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">v</font> = 
 <a href="vectsp10.html#K4" title="VECTSP10:func.4">zeroCoset</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_3_1"><span class="lab"><font color="Green" title="E41">C2</font></span></a>, <a class="ref" href="zmodul01.html#T63" onmouseover="rs('zmodul01/T63')" onmouseout="rh()">ZMODUL01:63</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:64_3_1"/>
<font color="Maroon" title="c11">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T10" onmouseover="rs('ranknull/T10')" onmouseout="rh()">RANKNULL:10</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>
<span class="kw">hence </span><a NAME="E2:64_3"/>
 <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T46" onmouseover="rs('zmodul01/T46')" onmouseout="rh()">ZMODUL01:46</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="E8:64"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
 <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:64_4"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> holds <br/><font color="Maroon" title="c10">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> implies <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_4_1"/><span class="lab"><font color="Green" title="E41">C2</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</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"> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:64_4_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_4_1"><span class="lab"><font color="Green" title="E41">C2</font></span></a>, <a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">yy</font> being    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c5">U1</font><span class="kw"> such that </span><br/><a NAME="E4:64_4_1"/><span class="lab"><font color="Green" title="E42">C8</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yy</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:64_4_1"/><span class="lab"><font color="Green" title="E43">C3</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_4_1"><span class="lab"><font color="Green" title="E42">C8</font></span></a>, <a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</a></span>;<br/>
<a NAME="E7:64_4_1"/>
<font color="Maroon" title="c13">z</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">z1</font>, <font color="Maroon" title="c15">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E9:64_4_1"/><span class="lab"><font color="Green" title="E44">C4</font></span>: 
( <font color="Maroon" title="c14">z1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c15">z2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">z2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T92" onmouseover="rs('zmodul01/T92')" onmouseout="rh()">ZMODUL01:92</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">zz1</font> = <font color="Maroon" title="c14">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64_4_1"><span class="lab"><font color="Green" title="E44">C4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">zzz1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:64_4_1"/><span class="lab"><font color="Green" title="E45">C6</font></span>: 
( <font color="Maroon" title="c16">zz1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">zzz1</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">zz1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">zzz1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c5">U1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/>
<a NAME="E13:64_4_1"/>
<font color="Maroon" title="c15">z2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64_4_1"><span class="lab"><font color="Green" title="E44">C4</font></span></a>, <a class="ref" href="zmodul01.html#T93" onmouseover="rs('zmodul01/T93')" onmouseout="rh()">ZMODUL01:93</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">zzz2</font> = <font color="Maroon" title="c15">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> ;<br/>
<a NAME="E15:64_4_1"/><span class="lab"><font color="Green" title="E46">C7</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">zzz1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">zzz2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64_4_1"><span class="lab"><font color="Green" title="E44">C4</font></span></a>, <a class="txt" href="#E12:64_4_1"><span class="lab"><font color="Green" title="E45">C6</font></span></a>, <a class="ref" href="zmodul01.html#T28" onmouseover="rs('zmodul01/T28')" onmouseout="rh()">ZMODUL01:28</a></span>;<br/>
<a NAME="E16:64_4_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">zz1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:64_4_1"><span class="lab"><font color="Green" title="E43">C3</font></span></a>, <a class="txt" href="#E9:64_4_1"><span class="lab"><font color="Green" title="E44">C4</font></span></a>, <a class="txt" href="#E12:64_4_1"><span class="lab"><font color="Green" title="E45">C6</font></span></a>, <a class="txt" href="#E15:64_4_1"><span class="lab"><font color="Green" title="E46">C7</font></span></a>, <a class="ref" href="zmodul01.html#T65" onmouseover="rs('zmodul01/T65')" onmouseout="rh()">ZMODUL01:65</a></span>;<br/>
<a NAME="E17:64_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">zz1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:64_4_1"/>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</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="E2:64_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">B1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T41" onmouseover="rs('funct_2/T41')" onmouseout="rh()">FUNCT_2:41</a></span>;<br/>
<a NAME="E3:64_4"/><span class="lab"><font color="Green" title="E42">B2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c10">f</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c10">f</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c10">f</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">W1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#D2" onmouseover="rs('ranknull/D2')" onmouseout="rh()">RANKNULL:def 2</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_4"><span class="lab"><font color="Green" title="E41">B1</font></span></a>, <a class="txt" href="#E3:64_4"><span class="lab"><font color="Green" title="E42">B2</font></span></a>, <a class="ref" href="relat_1.html#T113" onmouseover="rs('relat_1/T113')" onmouseout="rh()">RELAT_1:113</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:64_4"/>
 <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T47" onmouseover="rs('zmodul01/T47')" onmouseout="rh()">ZMODUL01:47</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>
<span class="kw">reconsider </span><font color="Maroon" title="c11">F</font> =  <a href="zmodul07.html#K6" title="ZMODUL07:func.6">Zdecom</a> <font color="Maroon" title="c10">f</font> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c6">U2</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E8:64"><span class="lab"><font color="Green" title="E40">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">FI</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>,<font color="Maroon" title="c5">U1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c6">U2</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E11:64"/><span class="lab"><font color="Green" title="E41">X1</font></span>: 
( <font color="Maroon" title="c12">FI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Maroon" title="c12">FI</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E8:64"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="zmodul07.html#D6" target="_self" onmouseover="rs('zmodul07/D6')" onmouseout="rh()">defdecom</a>, <a class="ref" href="zmodul06.html#T42" onmouseover="rs('zmodul06/T42')" onmouseout="rh()">ZMODUL06:42</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">FI</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="c12">FI</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<span class="kw">thus </span><a NAME="E12:64"/>
<font color="Maroon" title="c12">FI</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:64"><span class="lab"><font color="Green" title="E41">X1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
