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<b>let </b><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#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">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="c1">K</font>;<br/>



<div><i><font color="Green" title="E33">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V12" title="MOD_4:attr.12">antiautomorphism</a>
 ;<br/><a NAME="E2:53_1"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D17" target="_self" title="MOD_4:def.17">Def17</a></i>;<br/><a NAME="E3:53_1"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V4" title="MOD_4:attr.4">antimonomorphism</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D13" target="_self" title="MOD_4:def.13">Def13</a></i>;<br/><a NAME="E4:53_1"/><b>then </b>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D9" target="_self" title="MOD_4:def.9">Def9</a></i>;<br/><a NAME="E5:53_1"/><b>then </b>
( <font color="Maroon" title="c2">J</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> &amp; <font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V1" title="MOD_4:attr.1">antimultiplicative</a> &amp; <font color="Maroon" title="c2">J</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> )
 <b>by </b><i><a class="ref" href="mod_4.html#D7" target="_self" title="MOD_4:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E6:53_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Maroon" title="c2">J</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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> &amp; ( for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds <br/> ( <font color="Maroon" title="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">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="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span>)</span> &amp; <font color="Maroon" title="c2">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="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> ) ) )
 <b>by </b><i><a class="ref" href="mod_4.html#D6" target="_self" title="MOD_4:def.6">Def6</a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></i>;<br/><b>thus </b><a NAME="E7:53_1"/>
<font color="Maroon" title="c2">J</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E3:53_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="mod_4.html#D9" target="_self" title="MOD_4:def.9">Def9</a></i>;<br/><b>thus </b><a NAME="E8:53_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">J</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="c1">K</font>
 <b>by </b><i><a class="txt" href="mod_4.html#E2:53_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="mod_4.html#D13" target="_self" title="MOD_4:def.13">Def13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:53_2"/><i><font color="Green" title="E34">A4</font></i>: 
( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds  <font color="Maroon" title="c2">J</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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds  <font color="Maroon" title="c2">J</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="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"><font color="Maroon" title="c2">J</font> <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"><font color="Maroon" title="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) &amp; <font color="Maroon" title="c2">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="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> )
 <b>and </b><br/><a NAME="E2:53_2"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c2">J</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E3:53_2"/><i><font color="Green" title="E36">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">J</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="c1">K</font>
 ;<br/><a NAME="E4:53_2"/>
( <font color="Maroon" title="c2">J</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> &amp; <font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V1" title="MOD_4:attr.1">antimultiplicative</a> &amp; <font color="Maroon" title="c2">J</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> )
 <b>by </b><i><a class="txt" href="mod_4.html#E1:53_2"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="mod_4.html#D6" target="_self" title="MOD_4:def.6">Def6</a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></i>;<br/><a NAME="E5:53_2"/><b>then </b>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D7" target="_self" title="MOD_4:def.7">Def7</a></i>;<br/><a NAME="E6:53_2"/><b>then </b>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V4" title="MOD_4:attr.4">antimonomorphism</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E2:53_2"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="mod_4.html#D9" target="_self" title="MOD_4:def.9">Def9</a></i>;<br/><a NAME="E7:53_2"/><b>then </b>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V8" title="MOD_4:attr.8">antiisomorphism</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E3:53_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="mod_4.html#D13" target="_self" title="MOD_4:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E8:53_2"/>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V12" title="MOD_4:attr.12">antiautomorphism</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D17" target="_self" title="MOD_4:def.17">Def17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:53"/>
( <font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V12" title="MOD_4:attr.12">antiautomorphism</a> iff ( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds  <font color="Maroon" title="c2">J</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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> holds  <font color="Maroon" title="c2">J</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="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"><font color="Maroon" title="c2">J</font> <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"><font color="Maroon" title="c2">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) &amp; <font color="Maroon" title="c2">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="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> &amp; <font color="Maroon" title="c2">J</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">J</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="c1">K</font> ) )
 <b>by </b><i><a class="txt" href="mod_4.html#E1:53"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>


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