<?xml version="1.0"?>
<div class="add">

<span class="kw">thus </span><a NAME="E1:35_1_1"/>
 not  <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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"> (  <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a>  is  <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:35_1_1"/>
 <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</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">  <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a>  is  <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">xf</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">yf</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">zf</font> = <font color="Maroon" title="c6">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="anproj_9.html#K1" title="ANPROJ_9:func.1">EnsHomography3</a>  ;<br/>
<a NAME="E2:35_1_1_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">xf</font> <a href="anproj_9.html#K2" title="ANPROJ_9:func.2">(*)</a> <font color="Maroon" title="c8">yf</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_9.html#D3" target="_self" onmouseover="rs('anproj_9/D3')" onmouseout="rh()">Def02</a></span>;<br/>
<a NAME="E3:35_1_1_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">xf</font> <a href="anproj_9.html#K2" title="ANPROJ_9:func.2">(*)</a> <font color="Maroon" title="c8">yf</font></span>)</span> <a href="anproj_9.html#K2" title="ANPROJ_9:func.2">(*)</a> <font color="Maroon" title="c9">zf</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_9.html#D3" target="_self" onmouseover="rs('anproj_9/D3')" onmouseout="rh()">Def02</a>, <a class="txt" href="#E2:35_1_1_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/>
<a NAME="E4:35_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">yf</font> <a href="anproj_9.html#K2" title="ANPROJ_9:func.2">(*)</a> <font color="Maroon" title="c9">zf</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_9.html#D3" target="_self" onmouseover="rs('anproj_9/D3')" onmouseout="rh()">Def02</a></span>;<br/>
<a NAME="E5:35_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">xf</font> <a href="anproj_9.html#K2" title="ANPROJ_9:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">yf</font> <a href="anproj_9.html#K2" title="ANPROJ_9:func.2">(*)</a> <font color="Maroon" title="c9">zf</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_9.html#D3" target="_self" onmouseover="rs('anproj_9/D3')" onmouseout="rh()">Def02</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="anproj_9.html#T19" target="_self" onmouseover="rs('anproj_9/T19')" onmouseout="rh()">Ta1</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">take </span>
<font color="Maroon" title="c3">e</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="group_1.html#D2" onmouseover="rs('group_1/D2')" onmouseout="rh()">GROUP_1:def 2</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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a> holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  ex <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</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="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp;  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a> st <br/>( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:35_1_1"/>
( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="anproj_9.html#E30"><span class="lab"><font color="Green" title="E26">Lm1</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">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a> st <br/>( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<a NAME="E4:35_1_1"/>
<font color="Maroon" title="c4">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="anproj_9.html#K1" title="ANPROJ_9:func.1">EnsHomography3</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">N</font> being   <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a><span class="kw"> such that </span><br/><a NAME="E6:35_1_1"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c5">N</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">Ng</font> = <font color="Maroon" title="c5">N</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a>  as   <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> ;<br/>
<a NAME="E8:35_1_1"/>
 <a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c6">Ng</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="anproj_9.html#K1" title="ANPROJ_9:func.1">EnsHomography3</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> =  <a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c6">Ng</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="anproj_9.html#K4" title="ANPROJ_9:func.4">GroupHomography3</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">g</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="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c7">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<span class="kw">thus </span><a NAME="E10:35_1_1"/>
( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c7">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="anproj_9.html#E31"><span class="lab"><font color="Green" title="E27">Lm2</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>
