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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>, <font color="Olive" title="b2">U</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="b3">V1</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">U</font><br/>  for <font color="Olive" title="b5">f</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">U</font>  st <font color="Olive" title="b5">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b4">U1</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> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b3">V1</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">U</font>,<font color="Olive" title="b4">U1</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>, <font color="Maroon" title="c3">U</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">V1</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="b2">U1</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c3">U</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">U</font>  st <font color="Olive" title="b3">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">U1</font> holds <br/> ex <font color="Olive" title="b4">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> (<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">V1</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">U</font>,<font color="Olive" title="b2">U1</font>)</span>)</span> st <font color="Olive" title="b4">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="c4">V1</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">U</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">U</font>  st <font color="Olive" title="b2">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">U1</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> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">V1</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">U</font>,<font color="Olive" title="b1">U1</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">U</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="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">U</font>  st <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</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> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">V1</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">U</font>,<font color="Maroon" title="c5">U1</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">f</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">U</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">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</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> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">V1</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">U</font>,<font color="Maroon" title="c5">U1</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="E1:63"/><span class="lab"><font color="Green" title="E36">AS</font></span>: 
( <font color="Maroon" title="c6">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</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> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">V1</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">U</font>,<font color="Maroon" title="c5">U1</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">g</font> =  <a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">V1s</font> =  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c4">V1</font> as   <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="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T42" onmouseover="rs('zmodul01/T42')" onmouseout="rh()">ZMODUL01:42</a></span>;<br/>
<a NAME="E3:63"/><span class="lab"><font color="Green" title="E37">P2</font></span>: 
<span class="p1">(<span class="default"><a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <a href="zmodul07.html#K7" title="ZMODUL07:func.7">*</a> <font color="Maroon" title="c6">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E36">AS</font></span></a>, <a class="ref" href="funct_2.html#T27" onmouseover="rs('funct_2/T27')" onmouseout="rh()">FUNCT_2:27</a></span>;<br/>
<a NAME="E4:63"/><span class="lab"><font color="Green" title="E38">P5</font></span>: 
 <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c4">V1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c8">V1s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T37" target="_self" onmouseover="rs('zmodul07/T37')" onmouseout="rh()">ThStrict1</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#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <a href="zmodul07.html#K7" title="ZMODUL07:func.7">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  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#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T29" target="_self" onmouseover="rs('zmodul07/T29')" onmouseout="rh()">LMFirst3</a></span>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c5">U1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T31" target="_self" onmouseover="rs('zmodul07/T31')" onmouseout="rh()">LMFirst5</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">V1s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E36">AS</font></span></a></span>
;<br/>
<a NAME="E6:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">P3</font></span>: 
 <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <a href="zmodul07.html#K7" title="ZMODUL07:func.7">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">V1s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T45" onmouseover="rs('zmodul01/T45')" onmouseout="rh()">ZMODUL01:45</a></span>;<br/>
<span class="lab"><font color="Green" title="E40">P4</font></span>:  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <a href="zmodul07.html#K7" title="ZMODUL07:func.7">*</a> <font color="Maroon" title="c6">f</font></span>)</span> = 
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E37">P2</font></span></a>, <a class="ref" href="zmodul07.html#T30" target="_self" onmouseover="rs('zmodul07/T30')" onmouseout="rh()">LMFirst4</a></span>
<br/>.= 
 <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)

;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">F</font> =  <a href="zmodul07.html#K6" title="ZMODUL07:func.6">Zdecom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <a href="zmodul07.html#K7" title="ZMODUL07:func.7">*</a> <font color="Maroon" title="c6">f</font></span>)</span> 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="c2">V</font>,<font color="Maroon" title="c4">V1</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">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E39">P3</font></span></a>, <a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E38">P5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">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="c9">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E9:63"/>
 <a href="zmodul07.html#K6" title="ZMODUL07:func.6">Zdecom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zmodul07.html#K2" title="ZMODUL07:func.2">ZQMorph</a> (<font color="Maroon" title="c3">U</font>,<font color="Maroon" title="c5">U1</font>)</span>)</span> <a href="zmodul07.html#K7" title="ZMODUL07:func.7">*</a> <font color="Maroon" title="c6">f</font></span>)</span> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#D6" target="_self" onmouseover="rs('zmodul07/D6')" onmouseout="rh()">defdecom</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:63"/>
<font color="Maroon" title="c9">F</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="#E7:63"><span class="lab"><font color="Green" title="E40">P4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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