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

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b2">J</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">K</font>,<font color="Olive" title="b1">L</font> holds <br/> ( <font color="Olive" title="b2">J</font> is  <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a>  iff  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Olive" title="b2">J</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">J</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">L</font> holds <br/> ( <font color="Olive" title="b1">J</font> is  <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a>  iff  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Olive" title="b1">J</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">J</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">J</font> is  <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a>  iff  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">J9</font> =  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">L9</font> =  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">L</font>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>  implies <font color="Maroon" title="c3">J</font> is  <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:71_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c3">J</font> is  <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a> </span><br/><a NAME="E2:71_1"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:71_1"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="mod_4.html#D6" target="_self" title="MOD_4:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="mod_4.html#E5"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="p1">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">L</font></span>)</span>

;<br/><a NAME="E5:71_1"/><span class="kw">then </span>
 <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E6:71_1"/>
 <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="ringcat1.html#D2" title="RINGCAT1:def.2">RINGCAT1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:71"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
(  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp;  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp;  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ringcat1.html#D2" title="RINGCAT1:def.2">RINGCAT1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">J</font> is  <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a> </span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a>,<a class="ref" href="mod_4.html#D7" target="_self" title="MOD_4:def.7">MOD_4:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">J</font> is  <a href="mod_4.html#V1" title="MOD_4:attr.1">antimultiplicative</a>  &amp; <font color="Maroon" title="c3">J</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="mod_4.html#D6" target="_self" title="MOD_4:def.6">MOD_4:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">J</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="mod_4.html#E5"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">L</font>

; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
