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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Skew-Field</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="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font> is  <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</a>  iff <font color="Maroon" title="c1">K</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">J</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font>;<br/>
<div><span class="lab"><font color="Green" title="E27">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">now </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="c1">K</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a> implies  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font> is  <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c1">K</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</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="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font> is  <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</a> </span><br/><a NAME="E2:58_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font> holds  <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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"><span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</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="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</a></span>;<br/><a NAME="E3:58_1"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font>
 ;<br/><span class="kw">hence </span><a NAME="E4:58_1"/>
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font> is  <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="mod_4.html#T22" target="_self" onmouseover="rs('mod_4/T22')" onmouseout="rh()">Th22</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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_2"><span class="kw">now </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"> (  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font> is  <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</a>  implies <font color="Maroon" title="c1">K</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_2"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font> is  <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</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="c1">K</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a></span><br/><a NAME="E2:58_2"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E1:58_2_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 
;<br/>
<a NAME="E2:58_2_1"/>
( <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</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="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:58_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_2"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:58_2_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="mod_4.html#T22" target="_self" onmouseover="rs('mod_4/T22')" onmouseout="rh()">Th22</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="E3:58_2"/>
<font color="Maroon" title="c1">K</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</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="E3:58"/>
(  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">K</font> is  <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</a>  iff <font color="Maroon" title="c1">K</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E27">A1</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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