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<div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> implies <font color="Maroon" title="c5">X</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><br/><span class="kw">assume </span><a NAME="E1:7_1_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</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="c5">X</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><br/><a NAME="E2:7_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">R</font>,<font color="Olive" title="b1">x</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:7_1_1_1"/>
<font color="Maroon" title="c5">X</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="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T20" title="EQREL_1:th.20">EQREL_1:20</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">then </span><span class="kw">consider </span><font color="Maroon" title="c5">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:7_1_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">and </span><br/><a NAME="E4:7_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> holds <br/><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T111" title="FUNCT_1:th.111">FUNCT_1:111</a></span>;<br/>
<a NAME="E5:7_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">g</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">g</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:7_1_1_2"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E4:7_1_1_2"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:7_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E4:7_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E3:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E4:7_1_1_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:7_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E3:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></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">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> =  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> (<font color="Maroon" title="c3">R</font>,$1);<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,<span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:7_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c5">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E5:7_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> = <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> ($1,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $2</span>)</span>)</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">bR</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:7_1_1"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> holds  <font color="Maroon" title="c8">bR</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="stacks_1.html#S2" target="_self" title="STACKS_1:sch.2">STACKS_1:sch 2</a>();<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c8">bR</font>
; <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">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> &amp; <font color="Olive" title="b3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Maroon" title="c8">bR</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y1</font>)</span>)</span>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c11">y1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> &amp; <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font> implies <font color="Maroon" title="c8">bR</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E11:7_1_1"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E12:7_1_1"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">and </span><br/><a NAME="E13:7_1_1"/><span class="lab"><font color="Green" title="E11">A12</font></span>: 
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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">bR</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">x9</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">y9</font> = <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_1_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">y19</font> = <font color="Maroon" title="c11">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_1_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E13:7_1_1"><span class="lab"><font color="Green" title="E11">A12</font></span></a></span>;<br/>
<a NAME="E17:7_1_1"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
 ex <font color="Olive" title="b1">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c13">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">R</font>,<font color="Olive" title="b1">y2</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<a NAME="E18:7_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<a NAME="E19:7_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y9</font></span>)</span>,<font color="Maroon" title="c14">y19</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_1_1"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="txt" href="#E17:7_1_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="ref" href="eqrel_1.html#T22" title="EQREL_1:th.22">EQREL_1:22</a></span>;<br/>
<a NAME="E20:7_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">x9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y9</font></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">x9</font>,<font color="Maroon" title="c14">y19</font>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="stacks_1.html#D1" target="_self" title="STACKS_1:def.1">Def2</a></span>;<br/>
<a NAME="E21:7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A16</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">x9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y9</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">x9</font>,<font color="Maroon" title="c14">y19</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T19" title="EQREL_1:th.19">EQREL_1:19</a></span>;<br/>
<a NAME="E22:7_1_1"/><span class="lab"><font color="Green" title="E14">A17</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">x9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y9</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">x9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y9</font></span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>;<br/>
<a NAME="E23:7_1_1"/>
<font color="Maroon" title="c8">bR</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">x9</font>,<font color="Maroon" title="c13">y9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">x9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y9</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:7_1_1"><span class="lab"><font color="Green" title="E8">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:7_1_1"/>
<font color="Maroon" title="c8">bR</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:7_1_1"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="txt" href="#E22:7_1_1"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="eqrel_1.html#T23" title="EQREL_1:th.23">EQREL_1:23</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/>


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