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<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">P</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a> holds <br/><font color="Olive" title="b2">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c2">S</font></span><br/><b>let </b><font color="Maroon" title="c3">I</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a> holds <br/><font color="Olive" title="b1">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c2">S</font></span><br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a> implies <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c2">S</font> )</span><br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c4">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c2">S</font></span><br/>

<b>set </b><font color="Maroon" title="c5">Q</font> = <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font>;<br/>
<div><i><font color="Green" title="E26">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> implies <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> )</span><br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font></span><br/><b>then consider </b><font color="Maroon" title="c8">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c8">x'</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">R</font> &amp; <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">y'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:28_1"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c9">y'</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">R</font> &amp; <font color="Maroon" title="c9">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y'</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:28_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">x'</font> = <font color="Maroon" title="c8">x'</font>, <font color="Maroon" title="c11">y'</font> = <font color="Maroon" title="c9">y'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="hilbasis.html#E5:28_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/><a NAME="E7:28_1"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c10">x'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="hilbasis.html#E5:28_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a></i>;<br/><a NAME="E8:28_1"/>
( <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a> &amp; <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:28"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/><a NAME="E9:28_1"/><b>then </b>
<font color="Maroon" title="c4">P</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 <b>by </b><i><a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a></i>;<br/><a NAME="E10:28_1"/><b>then </b>
<font color="Maroon" title="c4">P</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
 <b>by </b><i><a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a></i>;<br/><a NAME="E11:28_1"/><b>then </b>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y'</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="hilbasis.html#E5:28_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E12:28_1"/>
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E7:28_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E27">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> implies <font color="Maroon" title="c6">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> )</span><br/><b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font></span><br/><b>then consider </b><font color="Maroon" title="c8">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green" title="E28">A8</font></i>: 
( <font color="Maroon" title="c8">x'</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">R</font> &amp; <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c8">x'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_2"><i><font color="Green" title="E28">A8</font></i></a></i>;<br/><a NAME="E5:28_2"/><i><font color="Green" title="E29">A9</font></i>: 
( <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a> &amp; <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:28"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/><a NAME="E6:28_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">P</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="c2">S</font>
 <b>by </b><i><a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">p'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:28_2"/><i><font color="Green" title="E30">A10</font></i>: 
( <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">p'</font> = <font color="Maroon" title="c10">p'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="hilbasis.html#E8:28_2"><i><font color="Green" title="E30">A10</font></i></a></i>;<br/><a NAME="E10:28_2"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c11">p'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a></i>;<br/><a NAME="E11:28_2"/>
<font color="Maroon" title="c4">P</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E5:28_2"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a></i>;<br/><a NAME="E12:28_2"/><b>then </b>
<font color="Maroon" title="c4">P</font> is <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>
 <b>by </b><i><a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a></i>;<br/><a NAME="E13:28_2"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">x'</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="hilbasis.html#E8:28_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/><b>hence </b><a NAME="E14:28_2"/>
<font color="Maroon" title="c6">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:28_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> implies <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> )</span><br/><b>assume </b><a NAME="E1:28_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font></span><br/><b>then consider </b><font color="Maroon" title="c8">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green" title="E28">A12</font></i>: 
( <font color="Maroon" title="c8">x'</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">R</font> &amp; <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c8">x'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_3"><i><font color="Green" title="E28">A12</font></i></a></i>;<br/><a NAME="E5:28_3"/><i><font color="Green" title="E29">A13</font></i>: 
( <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a> &amp; <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:28"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/><a NAME="E6:28_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">P</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="c2">S</font>
 <b>by </b><i><a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">p'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:28_3"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">p'</font> = <font color="Maroon" title="c10">p'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="hilbasis.html#E8:28_3"><i><font color="Green" title="E30">A14</font></i></a></i>;<br/><a NAME="E10:28_3"/><i><font color="Green" title="E31">A15</font></i>: 
<font color="Maroon" title="c9">x'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_3"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="ideal_1.html#D3" title="IDEAL_1:def.3">IDEAL_1:def 3</a></i>;<br/><a NAME="E11:28_3"/>
<font color="Maroon" title="c4">P</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E5:28_3"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a></i>;<br/><a NAME="E12:28_3"/><b>then </b>
<font color="Maroon" title="c4">P</font> is <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>
 <b>by </b><i><a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a></i>;<br/><a NAME="E13:28_3"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">p'</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:28_3"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="hilbasis.html#E8:28_3"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/><b>hence </b><a NAME="E14:28_3"/>
<font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:28_3"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:28"/>
<font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E2:28"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="hilbasis.html#E3:28"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a>, <a class="ref" href="ideal_1.html#D3" title="IDEAL_1:def.3">IDEAL_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
