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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> , <font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">I</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">R</font>,<font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">P</font> is <a href="quofield.html#V4">RingIsomorphism</a>
 ;<br/>

<b>set </b><font color="Maroon">Q</font> = <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font>;<br/>
<div><i><font color="Green">E45</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font> )
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon">y'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:28_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font>, <font color="Maroon">y'</font> = <font color="Maroon">y'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i>, </i>;<br/><a NAME="E7:28_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , <a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a></i>;<br/><a NAME="E8:28_1"/>
( <font color="Maroon">P</font> is <a href="quofield.html#V3">RingMonomorphism</a> &amp; <font color="Maroon">P</font> is <a href="quofield.html#V2">RingEpimorphism</a> )
 <b>by </b><i>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/><a NAME="E9:28_1"/><b>then </b>
<font color="Maroon">P</font> is <a href="quofield.html#V1">RingHomomorphism</a>
 <b>by </b><i><a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a></i>;<br/><a NAME="E10:28_1"/><b>then </b>
<font color="Maroon">P</font> is <a href="grcat_1.html#V1">additive</a>
 <b>by </b><i><a class="ref" href="quofield.html#D21">QUOFIELD:def 21</a></i>;<br/><a NAME="E11:28_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E12:28_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E50</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">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E5:28_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">P</font> is <a href="quofield.html#V3">RingMonomorphism</a> &amp; <font color="Maroon">P</font> is <a href="quofield.html#V2">RingEpimorphism</a> )
 <b>by </b><i>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/><a NAME="E6:28_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a></i>;<br/><b>then consider </b><font color="Maroon">p'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:28_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E10:28_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#D2">IDEAL_1:def 2</a></i>;<br/><a NAME="E11:28_2"/>
<font color="Maroon">P</font> is <a href="quofield.html#V1">RingHomomorphism</a>
 <b>by </b><i>, <a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a></i>;<br/><a NAME="E12:28_2"/><b>then </b>
<font color="Maroon">P</font> is <a href="group_6.html#V1">multiplicative</a>
 <b>by </b><i><a class="ref" href="quofield.html#D21">QUOFIELD:def 21</a></i>;<br/><a NAME="E13:28_2"/><b>then </b>
<font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/><b>hence </b><a NAME="E14:28_2"/>
<font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<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">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:28_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E5:28_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">P</font> is <a href="quofield.html#V3">RingMonomorphism</a> &amp; <font color="Maroon">P</font> is <a href="quofield.html#V2">RingEpimorphism</a> )
 <b>by </b><i>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/><a NAME="E6:28_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a></i>;<br/><b>then consider </b><font color="Maroon">p'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:28_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E10:28_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#D3">IDEAL_1:def 3</a></i>;<br/><a NAME="E11:28_3"/>
<font color="Maroon">P</font> is <a href="quofield.html#V1">RingHomomorphism</a>
 <b>by </b><i>, <a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a></i>;<br/><a NAME="E12:28_3"/><b>then </b>
<font color="Maroon">P</font> is <a href="group_6.html#V1">multiplicative</a>
 <b>by </b><i><a class="ref" href="quofield.html#D21">QUOFIELD:def 21</a></i>;<br/><a NAME="E13:28_3"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p'</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/><b>hence </b><a NAME="E14:28_3"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:28"/>
<font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a>, <a class="ref" href="ideal_1.html#D2">IDEAL_1:def 2</a>, <a class="ref" href="ideal_1.html#D3">IDEAL_1:def 3</a></i>;<br/>


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