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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">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="E41">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V11" title="MOD_4:attr.11">antiautomorphism</a>
 ;<br/><a NAME="E2:47_1"/><b>then </b><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V7" title="MOD_4:attr.7">antiisomorphism</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D16" target="_self" title="MOD_4:def.16">Def16</a></i>;<br/><a NAME="E3:47_1"/><b>then </b><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V3" title="MOD_4:attr.3">antimonomorphism</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D12" target="_self" title="MOD_4:def.12">Def12</a></i>;<br/><a NAME="E4:47_1"/><b>then </b>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V1" title="MOD_4:attr.1">antilinear</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D8" target="_self" title="MOD_4:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E5:47_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">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="rlvect_1.html#K2" title="RLVECT_1:func.2">+</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="rlvect_1.html#K2" title="RLVECT_1:func.2">+</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#NM3" title="VECTSP_1:NM.3">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="group_1.html#K1" title="GROUP_1:func.1">*</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="group_1.html#K1" title="GROUP_1: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="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#K2" title="GROUP_1:func.2">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#K2" title="GROUP_1:func.2">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></i>;<br/><b>thus </b><a NAME="E6:47_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#E3:47_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="mod_4.html#D8" target="_self" title="MOD_4:def.8">Def8</a></i>;<br/><b>thus </b><a NAME="E7:47_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#E2:47_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="mod_4.html#D12" target="_self" title="MOD_4:def.12">Def12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:47_2"/><i><font color="Green" title="E42">A4</font></i>: 
( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">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="rlvect_1.html#K2" title="RLVECT_1:func.2">+</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="rlvect_1.html#K2" title="RLVECT_1:func.2">+</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#NM3" title="VECTSP_1:NM.3">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="group_1.html#K1" title="GROUP_1: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="b2">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1: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="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#K2" title="GROUP_1:func.2">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#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">K</font> )
 <b>and </b><br/><a NAME="E2:47_2"/><i><font color="Green" title="E43">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="E3:47_2"/><i><font color="Green" title="E44">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="E4:47_2"/>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V1" title="MOD_4:attr.1">antilinear</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E1:47_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="mod_4.html#D6" target="_self" title="MOD_4:def.6">Def6</a></i>;<br/><a NAME="E5:47_2"/><b>then </b>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V3" title="MOD_4:attr.3">antimonomorphism</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E2:47_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="mod_4.html#D8" target="_self" title="MOD_4:def.8">Def8</a></i>;<br/><a NAME="E6:47_2"/><b>then </b>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V7" title="MOD_4:attr.7">antiisomorphism</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E3:47_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="mod_4.html#D12" target="_self" title="MOD_4:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E7:47_2"/>
<font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V11" title="MOD_4:attr.11">antiautomorphism</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D16" target="_self" title="MOD_4:def.16">Def16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47"/>
( <font color="Maroon" title="c2">J</font> is <a href="mod_4.html#V11" title="MOD_4:attr.11">antiautomorphism</a> iff ( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">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="rlvect_1.html#K2" title="RLVECT_1:func.2">+</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="rlvect_1.html#K2" title="RLVECT_1:func.2">+</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#NM3" title="VECTSP_1:NM.3">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="group_1.html#K1" title="GROUP_1: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="b2">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1: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="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#K2" title="GROUP_1:func.2">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#K2" title="GROUP_1:func.2">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#E1:47"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/>


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