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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">CI</font> being   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <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">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="b2">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  holds <br/><font color="Olive" title="b2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">CI</font> is   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">X9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">CI</font> be   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <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">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">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">CI</font> is   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">X9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="c4">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">CI</font> is   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">X9</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:62"/>
<font color="Maroon" title="c4">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="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">CI</font> is   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">X9</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">Kf</font> = <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">CI</font> as    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">X9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T47" target="_self" onmouseover="rs('bcialg_6/T47')" onmouseout="rh()">Th47</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">Kf</font> holds  <font color="Olive" title="b1">x9</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kf</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x9</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">Kf</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="c6">x9</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kf</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E2:62_1"/>
<font color="Maroon" title="c7">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="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E3:62_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CI</font>
 <span class="kw">and </span><br/><a NAME="E4:62_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c6">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">CI</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a></span>;<br/><a NAME="E6:62_1"/>
<font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="tarski.html#R2" title="TARSKI: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>
 ;<br/><a NAME="E7:62_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</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="E8:62_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">CI</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span></span>)</span></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="bcialg_1.html#D19" onmouseover="rs('bcialg_1/D19')" onmouseout="rh()">BCIALG_1:def 19</a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/><a NAME="E9:62_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">CI</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D13" onmouseover="rs('relat_1/D13')" onmouseout="rh()">RELAT_1:def 13</a></span>;<br/><a NAME="E10:62_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">CI</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D6" target="_self" onmouseover="rs('bcialg_6/D6')" onmouseout="rh()">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E11:62_1"/>
<font color="Maroon" title="c6">x9</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="bcialg_6.html#T35" target="_self" onmouseover="rs('bcialg_6/T35')" onmouseout="rh()">Th35</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>
<span class="kw">hence </span><a NAME="E4:62"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">CI</font> is   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D19" onmouseover="rs('bcialg_1/D19')" onmouseout="rh()">BCIALG_1:def 19</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/>


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