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<span class="kw">set </span><font color="Maroon" title="c6">IT</font> = <font color="Maroon" title="c5">f</font> <a href="fomodel3.html#K12" title="FOMODEL3:func.12">quotient</a> (<font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">F</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">P</font> =  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">E</font> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">EEE</font> = <font color="Maroon" title="c3">E</font> as   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">e</font>, <font color="Maroon" title="c10">f1</font>, <font color="Maroon" title="c11">f2</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c10">f1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="fomodel3.html#K12" title="FOMODEL3:func.12">quotient</a> (<font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">F</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="fomodel3.html#K12" title="FOMODEL3:func.12">quotient</a> (<font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">F</font>) implies <font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c10">f1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="fomodel3.html#K12" title="FOMODEL3:func.12">quotient</a> (<font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">F</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="fomodel3.html#K12" title="FOMODEL3:func.12">quotient</a> (<font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">F</font>) implies <font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">ee1</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">E</font>, <font color="Maroon" title="c13">ff1</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="c4">F</font><span class="kw"> such that </span><br/><a NAME="E3:47_1_1_1"/><span class="lab"><font color="Green" title="E18">C1</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c10">f1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">ee1</font>,<font color="Maroon" title="c13">ff1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">ee1</font> &amp; <font color="Olive" title="b2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">ff1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> ) )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c15">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:47_1_1_1"/><span class="lab"><font color="Green" title="E19">C3</font></span>: 
( <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">ee1</font> &amp; <font color="Maroon" title="c15">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">ff1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_1"><span class="lab"><font color="Green" title="E18">C1</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E6:47_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="fomodel3.html#K12" title="FOMODEL3:func.12">quotient</a> (<font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">F</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="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">ee2</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">E</font>, <font color="Maroon" title="c17">ff2</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="c4">F</font><span class="kw"> such that </span><br/><a NAME="E8:47_1_1_1"/><span class="lab"><font color="Green" title="E20">C2</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">ee2</font>,<font color="Maroon" title="c17">ff2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  ex <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">ee2</font> &amp; <font color="Olive" title="b2">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ff2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> ) )
 ;<br/><a NAME="E9:47_1_1_1"/><span class="lab"><font color="Green" title="E21">C9</font></span>: 
( <font color="Maroon" title="c12">ee1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e</font> &amp; <font color="Maroon" title="c16">ee2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e</font> &amp; <font color="Maroon" title="c13">ff1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font> &amp; <font color="Maroon" title="c17">ff2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_1"><span class="lab"><font color="Green" title="E18">C1</font></span></a>, <a class="txt" href="#E8:47_1_1_1"><span class="lab"><font color="Green" title="E20">C2</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">x2</font>, <font color="Maroon" title="c19">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:47_1_1_1"/><span class="lab"><font color="Green" title="E22">C4</font></span>: 
( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">ee2</font> &amp; <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ff2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">x2</font>,<font color="Maroon" title="c19">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_1_1"><span class="lab"><font color="Green" title="E20">C2</font></span></a></span>;<br/><a NAME="E12:47_1_1_1"/><span class="lab"><font color="Green" title="E23">C12</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c18">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel3.html#E21"><span class="lab"><font color="Green" title="E16">Lm3</font></span></a>, <a class="txt" href="#E5:47_1_1_1"><span class="lab"><font color="Green" title="E19">C3</font></span></a>, <a class="txt" href="#E11:47_1_1_1"><span class="lab"><font color="Green" title="E22">C4</font></span></a>, <a class="txt" href="#E9:47_1_1_1"><span class="lab"><font color="Green" title="E21">C9</font></span></a></span>;<br/><a NAME="E13:47_1_1_1"/>
( <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x2</font> &amp; <font color="Maroon" title="c15">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47_1_1_1"><span class="lab"><font color="Green" title="E22">C4</font></span></a>, <a class="txt" href="#E5:47_1_1_1"><span class="lab"><font color="Green" title="E19">C3</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><a NAME="E14:47_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y1</font>,<font color="Maroon" title="c19">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#D9" target="_self" title="FOMODEL3:def.9">DefCompatible</a>, <a class="txt" href="#E12:47_1_1_1"><span class="lab"><font color="Green" title="E23">C12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:47_1_1_1"/>
<font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1_1_1"><span class="lab"><font color="Green" title="E21">C9</font></span></a>, <a class="txt" href="#E5:47_1_1_1"><span class="lab"><font color="Green" title="E19">C3</font></span></a>, <a class="txt" href="#E11:47_1_1_1"><span class="lab"><font color="Green" title="E22">C4</font></span></a>, <a class="txt" href="fomodel3.html#E22"><span class="lab"><font color="Green" title="E17">Lm10</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">hence </span><a NAME="E4:47_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="fomodel3.html#K12" title="FOMODEL3:func.12">quotient</a> (<font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">F</font>) holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="funct_1.html#V1" title="FUNCT_1:attr.1">Function-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D1" title="FUNCT_1:def.1">FUNCT_1:def 1</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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