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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K9" title="QUOFIELD:func.9">Quot.</a> <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K9" title="QUOFIELD:func.9">Quot.</a> <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">v</font> implies <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E2:19"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:19"/>
<font color="Maroon" title="c2">u</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font></span><br/>

<a NAME="E4:19"/><span class="kw">then </span>
<font color="Maroon" title="c2">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:19"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font>
 <span class="kw">and </span><br/><a NAME="E7:19"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E9:19"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c5">w</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>;<br/>
<a NAME="E11:19"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:19"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></span>;<br/>
<a NAME="E12:19"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 for <font color="Olive" title="b1">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c4">x</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c6">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c4">x</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<a NAME="E1:19_1"/>
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
 
;<br/>
<a NAME="E2:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T7" title="GCD_1:th.7">GCD_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:19_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c6">y</font></span><br/>

<a NAME="E4:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="quofield.html#T7" target="_self" title="QUOFIELD:th.7">Th8</a></span>;<br/>
<a NAME="E5:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></span>;<br/>
<a NAME="E6:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T7" title="GCD_1:th.7">GCD_1:7</a></span>;<br/>
<a NAME="E7:19_1"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="gcd_1.html#D4" title="GCD_1:def.4">GCD_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E6:19_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="gcd_1.html#T11" title="GCD_1:th.11">GCD_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:19"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="gcd_1.html#T11" title="GCD_1:th.11">GCD_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="gcd_1.html#T9" title="GCD_1:th.9">GCD_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:19_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></span>; <a class="txt" 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>
<a NAME="E13:19"/><span class="lab"><font color="Green" title="E19">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></span>;<br/>
<a NAME="E14:19"/>
 for <font color="Olive" title="b1">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c6">y</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c4">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<a NAME="E1:19_2"/>
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
 
;<br/>
<a NAME="E2:19_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A13</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T7" title="GCD_1:th.7">GCD_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:19_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E4:19_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:19"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="quofield.html#T7" target="_self" title="QUOFIELD:th.7">Th8</a></span>;<br/>
<a NAME="E5:19_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A15</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></span>;<br/>
<a NAME="E6:19_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A16</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T7" title="GCD_1:th.7">GCD_1:7</a></span>;<br/>
<a NAME="E7:19_2"/><span class="lab"><font color="Green" title="E24">A17</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E21">A14</font></span></a>, <a class="txt" href="#E5:19_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="gcd_1.html#D4" title="GCD_1:def.4">GCD_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_2"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E6:19_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="txt" href="#E7:19_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="ref" href="gcd_1.html#T11" title="GCD_1:th.11">GCD_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:19"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2"><span class="lab"><font color="Green" title="E20">A13</font></span></a>, <a class="txt" href="#E7:19_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="ref" href="gcd_1.html#T11" title="GCD_1:th.11">GCD_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="ref" href="gcd_1.html#T9" title="GCD_1:th.9">GCD_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:19_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></span>; <a class="txt" 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="E15:19"/>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E9:19"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E12:19"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="subset_1.html#T3" title="SUBSET_1:th.3">SUBSET_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
