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<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <font color="Maroon" title="c1">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">RX</font> =  <a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B</font> =  <a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font>;<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D19" onmouseover="rs('vectsp_1/D19')" onmouseout="rh()">VECTSP_1:def 19</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="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp;  <a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp;  <a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</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="c7">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y</font></span>)</span></span><br/><a NAME="E1:30_1_1_1"/><span class="lab"><font color="Green" title="E16">B</font></span>: 
( <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font> &amp; <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D2" target="_self" onmouseover="rs('field_5/D2')" onmouseout="rh()">defch</a></span>;<br/><a NAME="E2:30_1_1_1"/><span class="lab"><font color="Green" title="E17">A</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">fx</font> = <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">fy</font> = <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1"><span class="lab"><font color="Green" title="E16">B</font></span></a>, <a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span> = 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#D8" onmouseover="rs('ring_4/D8')" onmouseout="rh()">RING_4:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c9">fx</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">fy</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1"><span class="lab"><font color="Green" title="E17">A</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1"><span class="lab"><font color="Green" title="E16">B</font></span></a>, <a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T17" onmouseover="rs('ring_4/T17')" onmouseout="rh()">RING_4:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</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="c7">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D2" target="_self" onmouseover="rs('field_5/D2')" onmouseout="rh()">defch</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><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</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="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp;  <a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</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="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font></span>)</span></span><br/><a NAME="E1:30_1_1_2"/><span class="lab"><font color="Green" title="E16">B</font></span>: 
( <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font> &amp; <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D2" target="_self" onmouseover="rs('field_5/D2')" onmouseout="rh()">defch</a></span>;<br/><a NAME="E2:30_1_1_2"/><span class="lab"><font color="Green" title="E17">A</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">fx</font> = <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">fy</font> = <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_2"><span class="lab"><font color="Green" title="E16">B</font></span></a>, <a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">p1</font> = <font color="Maroon" title="c2">p</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font> ;<br/><a NAME="E5:30_1_1_2"/><span class="lab"><font color="Green" title="E18">C</font></span>: 
(  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#D4" onmouseover="rs('ring_4/D4')" onmouseout="rh()">RING_4:def 4</a>, <a class="ref" href="ratfunc1.html#D2" onmouseover="rs('ratfunc1/D2')" onmouseout="rh()">RATFUNC1:def 2</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ring_4.html#K5" title="RING_4:func.5">poly_mult_mod</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#D8" onmouseover="rs('ring_4/D8')" onmouseout="rh()">RING_4:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ring_4.html#K5" title="RING_4:func.5">poly_mult_mod</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">fx</font>,<font color="Maroon" title="c10">fy</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_2"><span class="lab"><font color="Green" title="E17">A</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_2"><span class="lab"><font color="Green" title="E16">B</font></span></a>, <a class="ref" href="ring_4.html#D7" onmouseover="rs('ring_4/D7')" onmouseout="rh()">RING_4:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T18" onmouseover="rs('ring_4/T18')" onmouseout="rh()">RING_4:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_1_2"><span class="lab"><font color="Green" title="E18">C</font></span></a>, <a class="ref" href="ring_4.html#T2" onmouseover="rs('ring_4/T2')" onmouseout="rh()">RING_4:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</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="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D2" target="_self" onmouseover="rs('field_5/D2')" onmouseout="rh()">defch</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">thus </span><span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</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">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D2" target="_self" onmouseover="rs('field_5/D2')" onmouseout="rh()">defch</a></span>
<br/>.= 
 <a href="polynom3.html#K9" title="POLYNOM3:func.9">1_.</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T14" onmouseover="rs('ring_4/T14')" onmouseout="rh()">RING_4:14</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#D8" onmouseover="rs('ring_4/D8')" onmouseout="rh()">RING_4:def 8</a></span>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="group_1.html#D13" onmouseover="rs('group_1/D13')" onmouseout="rh()">GROUP_1:def 13</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="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_4"><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">  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> &amp; <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> &amp; <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font> implies <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:30_1_1_4"/><span class="lab"><font color="Green" title="E16">AS</font></span>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> &amp; <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</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="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c8">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> ;<br/><font color="Maroon" title="c9">a</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font> = 
<span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_4"><span class="lab"><font color="Green" title="E16">AS</font></span></a>, <a class="ref" href="field_5.html#D2" target="_self" onmouseover="rs('field_5/D2')" onmouseout="rh()">defch</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D2" target="_self" onmouseover="rs('field_5/D2')" onmouseout="rh()">defch</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:30_1_1_4"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T19" onmouseover="rs('ring_4/T19')" onmouseout="rh()">RING_4:19</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="E5:30_1_1"/>
 <a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T19" onmouseover="rs('funct_2/T19')" onmouseout="rh()">FUNCT_2:19</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>
