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<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K4" title="BCIALG_6:func.4">Ker</a> <font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  holds <br/><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>,<font color="Maroon" title="c2">X9</font> <a href="bcialg_6.html#R1" title="BCIALG_6:pred.1">are_isomorphic</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K4" title="BCIALG_6:func.4">Ker</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  holds <br/><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>,<font color="Maroon" title="c2">X9</font> <a href="bcialg_6.html#R1" title="BCIALG_6:pred.1">are_isomorphic</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K4" title="BCIALG_6:func.4">Ker</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  holds <br/><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>,<font color="Maroon" title="c2">X9</font> <a href="bcialg_6.html#R1" title="BCIALG_6:pred.1">are_isomorphic</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <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>,<font color="Maroon" title="c2">X9</font> <a href="bcialg_6.html#R1" title="BCIALG_6:pred.1">are_isomorphic</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E2:72"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>,<font color="Maroon" title="c2">X9</font> <a href="bcialg_6.html#R1" title="BCIALG_6:pred.1">are_isomorphic</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E4:72"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:72"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c6">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="#E1:72"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="bcialg_6.html#T50" target="_self" onmouseover="rs('bcialg_6/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<a NAME="E6:72"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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="c2">X9</font> holds <br/> ex <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">z</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 <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> &amp; <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="c2">X9</font> implies  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">z</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 <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> &amp; <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_1"/>
<font color="Maroon" title="c7">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="c2">X9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">z</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 <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> &amp; <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<a NAME="E2:72_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">y</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> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:72_1"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c8">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">and </span><br/><a NAME="E5:72_1"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c7">y</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="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">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="c8">x</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K5" title="BCIALG_6:func.5">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="c8">x</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 <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> &amp; <font color="Maroon" title="c6">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#K5" title="BCIALG_6:func.5">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="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<a NAME="E6:72_1"/><span class="lab"><font color="Green" title="E45">A7</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#K5" title="BCIALG_6:func.5">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:72_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">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="c8">x</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c4">RI</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:72_1"/>
( <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">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="c8">x</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 <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> &amp; <font color="Maroon" title="c6">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#K5" title="BCIALG_6:func.5">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="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E4:72_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E5:72_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E6:72_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E7:72"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">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 <font color="Maroon" title="c2">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T10" onmouseover="rs('funct_2/T10')" onmouseout="rh()">FUNCT_2:10</a></span>;<br/>
<a NAME="E8:72"/><span class="kw">then </span>
<font color="Maroon" title="c6">h</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:72"/>
<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>,<font color="Maroon" title="c2">X9</font> <a href="bcialg_6.html#R1" title="BCIALG_6:pred.1">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
