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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">X9</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</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">I</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">RI</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">I</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X9</font>  st <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Olive" title="b3">f</font> holds <br/> ex <font color="Olive" title="b4">h</font> being   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b2">RI</font></span>)</span>,<font color="Maroon" title="c2">X9</font> st <br/>( <font color="Olive" title="b3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Olive" title="b2">RI</font></span>)</span> &amp; <font color="Olive" title="b4">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">I</font> be    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</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">RI</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">I</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X9</font>  st <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b3">h</font> being   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b1">RI</font></span>)</span>,<font color="Maroon" title="c2">X9</font> st <br/>( <font color="Olive" title="b2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Olive" title="b1">RI</font></span>)</span> &amp; <font color="Olive" title="b3">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">RI</font> be    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">I</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">f</font> being   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X9</font>  st <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Olive" title="b1">f</font> holds <br/> ex <font color="Olive" title="b2">h</font> being   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>,<font color="Maroon" title="c2">X9</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <font color="Olive" title="b2">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X9</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="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Maroon" title="c5">f</font> implies  ex <font color="Olive" title="b1">h</font> being   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>,<font color="Maroon" title="c2">X9</font> st <br/>( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <font color="Olive" title="b1">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">J</font> = <font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st $1 <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="c4">RI</font>,<font color="Olive" title="b1">a</font>) holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>;<br/>
<a NAME="E1:69"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:69"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>,<font color="Maroon" title="c2">X9</font> st <br/>( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <font color="Olive" title="b1">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<a NAME="E3:69"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">X9</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">X9</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:69_1"/>
<font color="Maroon" title="c7">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="c4">RI</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E4:69_1"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c7">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="c4">RI</font>,<font color="Maroon" title="c8">y</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">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">X9</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">t</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">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">t</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">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="c7">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="c4">RI</font>,<font color="Maroon" title="c11">a</font>) implies <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E7:69_1"/>
<font color="Maroon" title="c7">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="c4">RI</font>,<font color="Maroon" title="c11">a</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">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span><br/>

<a NAME="E8:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RI</font>,<font color="Maroon" title="c11">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="eqrel_1.html#T23" title="EQREL_1:th.23">EQREL_1:23</a></span>;<br/>
<a NAME="E9:69_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c9">y</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">RI</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" title="EQREL_1:th.18">EQREL_1:18</a></span>;<br/>
<a NAME="E10:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="bcialg_2.html#D12" title="BCIALG_2:def.12">BCIALG_2:def 12</a></span>;<br/>
<a NAME="E11:69_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c9">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X9</font> )
 
;<br/>
<a NAME="E12:69_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D6" target="_self" title="BCIALG_6:def.6">Def6</a></span>;<br/>
<a NAME="E13:69_1"/>
<font color="Maroon" title="c11">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E9:69_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="bcialg_2.html#D12" title="BCIALG_2:def.12">BCIALG_2:def 12</a></span>;<br/>
<a NAME="E14:69_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c11">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X9</font> )
 
;<br/>
<a NAME="E15:69_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D6" target="_self" title="BCIALG_6:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:69_1"/>
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="bcialg_1.html#D7" title="BCIALG_1:def.7">BCIALG_1:def 7</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">consider </span><font color="Maroon" title="c7">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>,<font color="Maroon" title="c2">X9</font><span class="kw"> such that </span><br/><a NAME="E5:69"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E41">A3</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c9">b</font></span>)</span></span><br/><a NAME="E1:69_2"/>
<font color="Maroon" title="c8">a</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="c4">RI</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:69_2"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c10">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E4:69_2"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c8">a</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="c4">RI</font>,<font color="Maroon" title="c10">a1</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="E5:69_2"/>
<font color="Maroon" title="c9">b</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="c4">RI</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:69_2"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c11">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E8:69_2"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<font color="Maroon" title="c9">b</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="c4">RI</font>,<font color="Maroon" title="c11">b1</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">reconsider </span><font color="Maroon" title="c12">a1</font> = <font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c13">b1</font> = <font color="Maroon" title="c11">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_2"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E7:69_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a></span>;<br/><a NAME="E10:69_2"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c9">b</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="c4">RI</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">b1</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_2"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E8:69_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="bcialg_2.html#D17" title="BCIALG_2:def.17">BCIALG_2:def 17</a></span>;<br/><a NAME="E11:69_2"/><span class="lab"><font color="Green" title="E48">A14</font></span>: 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E8:69_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a></span>;<br/><a NAME="E12:69_2"/>
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E4:69_2"><span class="lab"><font color="Green" title="E44">A10</font></span></a></span>;<br/><a NAME="E13:69_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:69_2"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="ref" href="bcialg_6.html#D6" target="_self" title="BCIALG_6:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E14:69_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c9">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E10:69_2"><span class="lab"><font color="Green" title="E47">A13</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c7">h</font> as   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>,<font color="Maroon" title="c2">X9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D6" target="_self" title="BCIALG_6:def.6">Def6</a></span>;<br/>
<a NAME="E8:69"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c8">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">y2</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="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c8">h</font> or  not <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c8">h</font> or  not <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y2</font> or <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69_3"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> )
 <span class="kw">and </span><br/><a NAME="E2:69_3"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y2</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="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">S</font> = <font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c12">T</font> = <font color="Maroon" title="c10">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_3"><span class="lab"><font color="Green" title="E44">A16</font></span></a></span>;<br/>
<a NAME="E4:69_3"/>
<font color="Maroon" title="c11">S</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="c4">RI</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:69_3"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
<font color="Maroon" title="c13">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E7:69_3"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
<font color="Maroon" title="c11">S</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="c4">RI</font>,<font color="Maroon" title="c13">a1</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="E8:69_3"/>
<font color="Maroon" title="c12">T</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="c4">RI</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:69_3"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<font color="Maroon" title="c14">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E11:69_3"/><span class="lab"><font color="Green" title="E49">A21</font></span>: 
<font color="Maroon" title="c12">T</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="c4">RI</font>,<font color="Maroon" title="c14">b1</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">reconsider </span><font color="Maroon" title="c15">a1</font> = <font color="Maroon" title="c13">a1</font>, <font color="Maroon" title="c16">b1</font> = <font color="Maroon" title="c14">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_3"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E10:69_3"><span class="lab"><font color="Green" title="E48">A20</font></span></a></span>;<br/>
<a NAME="E13:69_3"/><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E11:69_3"><span class="lab"><font color="Green" title="E49">A21</font></span></a></span>;<br/>
<a NAME="E14:69_3"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E7:69_3"><span class="lab"><font color="Green" title="E47">A19</font></span></a></span>;<br/>
<a NAME="E15:69_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b1</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_3"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E13:69_3"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></span>;<br/>
<a NAME="E16:69_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c15">a1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D6" target="_self" title="BCIALG_6:def.6">Def6</a></span>;<br/>
<a NAME="E17:69_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A24</font></span>: 
<font color="Maroon" title="c16">b1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c15">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Maroon" title="c5">f</font>
 
;<br/>
<a NAME="E18:69_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a1</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_3"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E14:69_3"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="txt" href="#E13:69_3"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></span>;<br/>
<a NAME="E19:69_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c16">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D6" target="_self" title="BCIALG_6:def.6">Def6</a></span>;<br/>
<a NAME="E20:69_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c16">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">Ker</a> <font color="Maroon" title="c5">f</font>
 
;<br/>
<a NAME="E21:69_3"/><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">a1</font>,<font color="Maroon" title="c16">b1</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">RI</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E17:69_3"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="ref" href="bcialg_2.html#D12" title="BCIALG_2:def.12">BCIALG_2:def 12</a></span>;<br/>
<a NAME="E22:69_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RI</font>,<font color="Maroon" title="c15">a1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" title="EQREL_1:th.18">EQREL_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:69_3"/>
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_3"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="txt" href="#E11:69_3"><span class="lab"><font color="Green" title="E49">A21</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/>


</div><span class="kw">end;</span></div>
<a NAME="E9:69"/><span class="lab"><font color="Green" title="E44">A25</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E45">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> implies ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> ) ) &amp; ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:69_4"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> implies ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:69_4_1"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> )</span><br/>

<span class="kw">hence </span><a NAME="E2:69_4_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E39">A1</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"> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font></span><br/>

<a NAME="E3:69_4_1"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RI</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:69_4_1"/>
<span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:69_4_1"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:69_4_1"/>
<span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_4_1"><span class="lab"><font color="Green" title="E47">A28</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">assume </span><span class="kw">that </span><br/><a NAME="E2:69_4"/><span class="lab"><font color="Green" title="E46">A29</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:69_4"/>
<span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">h</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font></span><br/><span class="kw">thus </span><a NAME="E4:69_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="txt" href="#E2:69_4"><span class="lab"><font color="Green" title="E46">A29</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">take </span>
<font color="Maroon" title="c8">h</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">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <font color="Maroon" title="c8">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:69_5"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:69_5"/>
<span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</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="c4">RI</font>,<font color="Maroon" title="c10">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D10" target="_self" title="BCIALG_6:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E4:69_5"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E42">A8</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="E12:69"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span> &amp; <font color="Maroon" title="c8">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E10:69"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="ref" href="funct_1.html#T10" title="FUNCT_1:th.10">FUNCT_1:10</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>
