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<span class="kw">let </span><font color="Maroon" title="c1">X</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">G</font> being    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">K</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="b3">RK</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="b2">K</font><br/>  for <font color="Olive" title="b4">K1</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Olive" title="b1">G</font>,<font color="Olive" title="b3">RK</font>)<br/>  for <font color="Olive" title="b5">RK1</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Olive" title="b1">G</font>,<font color="Olive" title="b3">RK</font>),<font color="Olive" title="b4">K1</font><br/>  for <font color="Olive" title="b6">I</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Olive" title="b1">G</font><br/>  for <font color="Olive" title="b7">RI</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Olive" title="b1">G</font>,<font color="Olive" title="b6">I</font>  st <font color="Olive" title="b5">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">RK</font> &amp; <font color="Olive" title="b6">I</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="Olive" title="b1">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> holds <br/><font color="Olive" title="b1">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b7">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Olive" title="b1">G</font>,<font color="Olive" title="b3">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b5">RK1</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="c2">G</font> be    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</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">K</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">RK</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">K</font><br/>  for <font color="Olive" title="b3">K1</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Olive" title="b2">RK</font>)<br/>  for <font color="Olive" title="b4">RK1</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Olive" title="b2">RK</font>),<font color="Olive" title="b3">K1</font><br/>  for <font color="Olive" title="b5">I</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b6">RI</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c2">G</font>,<font color="Olive" title="b5">I</font>  st <font color="Olive" title="b4">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">RK</font> &amp; <font color="Olive" title="b5">I</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">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b6">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Olive" title="b2">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b4">RK1</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">K</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">RK</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">K</font><br/>  for <font color="Olive" title="b2">K1</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Olive" title="b1">RK</font>)<br/>  for <font color="Olive" title="b3">RK1</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Olive" title="b1">RK</font>),<font color="Olive" title="b2">K1</font><br/>  for <font color="Olive" title="b4">I</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b5">RI</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c2">G</font>,<font color="Olive" title="b4">I</font>  st <font color="Olive" title="b3">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">RK</font> &amp; <font color="Olive" title="b4">I</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">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b5">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Olive" title="b1">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b3">RK1</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">RK</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">K</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">K1</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)<br/>  for <font color="Olive" title="b2">RK1</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>),<font color="Olive" title="b1">K1</font><br/>  for <font color="Olive" title="b3">I</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b4">RI</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c2">G</font>,<font color="Olive" title="b3">I</font>  st <font color="Olive" title="b2">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">RK</font> &amp; <font color="Olive" title="b3">I</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">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b4">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b2">RK1</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">K1</font> be    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</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">RK1</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>),<font color="Maroon" title="c5">K1</font><br/>  for <font color="Olive" title="b2">I</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b3">RI</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c2">G</font>,<font color="Olive" title="b2">I</font>  st <font color="Olive" title="b1">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">RK</font> &amp; <font color="Olive" title="b2">I</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">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b3">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b1">RK1</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="c6">RK1</font> be    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>),<font color="Maroon" title="c5">K1</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">I</font> being    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">G</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="c2">G</font>,<font color="Olive" title="b1">I</font>  st <font color="Maroon" title="c6">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">RK</font> &amp; <font color="Olive" title="b1">I</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">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b2">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</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="c7">I</font> be    <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c2">G</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="c2">G</font>,<font color="Maroon" title="c7">I</font>  st <font color="Maroon" title="c6">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">RK</font> &amp; <font color="Maroon" title="c7">I</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">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font> holds <br/><font color="Maroon" title="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Olive" title="b1">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</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="c8">RI</font> be    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c7">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"> ( <font color="Maroon" title="c6">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">RK</font> &amp; <font color="Maroon" title="c7">I</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">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</font> implies <font color="Maroon" title="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c8">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</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="E1:84"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c6">RK1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">RK</font>
 <span class="kw">and </span><br/><a NAME="E2:84"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c7">I</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">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">K</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="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c8">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font> <a href="bcialg_6.html#R1" title="BCIALG_6:pred.1">are_isomorphic</a> </span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$2 <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="c6">RK1</font>,$1);<br/>
<a NAME="E3:84"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">xx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c9">xx</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">G</font> or <font color="Maroon" title="c9">xx</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"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:84_1"/>
<font color="Maroon" title="c9">xx</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">G</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="c9">xx</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"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">xx</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> ;<br/>
<a NAME="E3:84_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="bcialg_1.html#D10" onmouseover="rs('bcialg_1/D10')" onmouseout="rh()">BCIALG_1:def 10</a></span>;<br/>
<a NAME="E4:84_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c10">x</font> <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="E5:84_1"/><span class="kw">then </span>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Maroon" title="c10">x</font>) <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 <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">RK</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" onmouseover="rs('eqrel_1/D3')" onmouseout="rh()">EQREL_1:def 3</a></span>;<br/>
<a NAME="E6:84_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A5</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Maroon" title="c10">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Olive" title="b1">a</font>)</span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> :  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Olive" title="b1">a</font>) <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 <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">RK</font></span>)</span> </span> } </span> 
 
;<br/>
<a NAME="E7:84_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c10">x</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">RK</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T5" onmouseover="rs('eqrel_1/T5')" onmouseout="rh()">EQREL_1:5</a></span>;<br/>
<a NAME="E8:84_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Maroon" title="c10">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:84_1"/>
<font color="Maroon" title="c9">xx</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"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</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="E4:84"/><span class="lab"><font color="Green" title="E46">A6</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 <font color="Maroon" title="c2">G</font>  ex <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span> 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</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">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">y</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c6">RK1</font>,<font color="Maroon" title="c9">x</font>);<br/>
<a NAME="E1:84_2"/>
<font color="Maroon" title="c9">x</font> <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="c2">G</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c6">RK1</font>,<font color="Maroon" title="c9">x</font>) as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="eqrel_1.html#D3" onmouseover="rs('eqrel_1/D3')" onmouseout="rh()">EQREL_1:def 3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">y</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">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:84_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:84"/><span class="lab"><font color="Green" title="E47">A7</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 <font color="Maroon" title="c2">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">f</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" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E46">A6</font></span></a>);<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_3"><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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> holds  <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</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="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font></span>)</span></span><br/><a NAME="E1:84_3"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="bcialg_1.html#D10" onmouseover="rs('bcialg_1/D10')" onmouseout="rh()">BCIALG_1:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">xa</font> = <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c13">xb</font> = <font color="Maroon" title="c11">b</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="E3:84_3"/>
( <font color="Maroon" title="c10">a</font> <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="c2">G</font> &amp; <font color="Maroon" title="c11">b</font> <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="c2">G</font> )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">Wa</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c6">RK1</font>,<font color="Maroon" title="c10">a</font>), <font color="Maroon" title="c15">Wb</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c6">RK1</font>,<font color="Maroon" title="c11">b</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c6">RK1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="eqrel_1.html#D3" onmouseover="rs('eqrel_1/D3')" onmouseout="rh()">EQREL_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">a1</font> = <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c17">b1</font> = <font color="Maroon" title="c11">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E45">A3</font></span></a></span>;<br/><a NAME="E6:84_3"/>
( <font color="Maroon" title="c14">Wa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c15">Wb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>;<br/><a NAME="E7:84_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <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="c6">RK1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c17">b1</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#D17" onmouseover="rs('bcialg_2/D17')" onmouseout="rh()">BCIALG_2:def 17</a></span>;<br/><a NAME="E8:84_3"/>
 <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) is    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T56" target="_self" onmouseover="rs('bcialg_6/T56')" onmouseout="rh()">Th55</a></span>;<br/><a NAME="E9:84_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">xa</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">xb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c17">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T34" target="_self" onmouseover="rs('bcialg_6/T34')" onmouseout="rh()">Th34</a></span>;<br/><a NAME="E10:84_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <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="c6">RK1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">b</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:84_3"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="bcialg_6.html#T34" target="_self" onmouseover="rs('bcialg_6/T34')" onmouseout="rh()">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E11:84_3"/>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E47">A7</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><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="bcialg_6.html#NM1" title="BCIALG_6:NM.1">BCI-homomorphism</a> of <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span> <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/>
<a NAME="E9:84"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">I</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c11">X9</font> = <span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font>;<br/>
<span class="kw">thus </span><a NAME="E1:84_4"/>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c10">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">I</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c10">f</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c10">f</font> or <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">I</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:84_4_1"/>
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c10">f</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="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">I</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E3:84_4_1"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">and </span><br/><a NAME="E4:84_4_1"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span>
 ;<br/>
<a NAME="E5:84_4_1"/>
( <font color="Maroon" title="c13">x</font> <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="c2">G</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="bcialg_1.html#D10" onmouseover="rs('bcialg_1/D10')" onmouseout="rh()">BCIALG_1:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">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="E7:84_4_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Maroon" title="c14">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">RK</font>,<span class="p1">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E4:84_4_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a></span>;<br/>
<a NAME="E8:84_4_1"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Maroon" title="c14">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</a></span>;<br/>
<a NAME="E9:84_4_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<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><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">RK</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<a NAME="E10:84_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A12</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#D12" onmouseover="rs('bcialg_2/D12')" onmouseout="rh()">BCIALG_2:def 12</a></span>;<br/>
<a NAME="E11:84_4_1"/>
<font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:84_4_1"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="bcialg_1.html#D18" onmouseover="rs('bcialg_1/D18')" onmouseout="rh()">BCIALG_1:def 18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:84_4_1"/>
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E3:84_4_1"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</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">let </span><font color="Maroon" title="c12">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">I</font> or <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c10">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:84_4"/><span class="lab"><font color="Green" title="E49">A13</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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"> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c10">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">h</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="#E2:84"><span class="lab"><font color="Green" title="E44">A2</font></span></a></span>;<br/>
<a NAME="E4:84_4"/>
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E2:84_4"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:84_4"/><span class="kw">then </span>
( <font color="Maroon" title="c13">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c13">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">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T2" onmouseover="rs('bcialg_1/T2')" onmouseout="rh()">BCIALG_1:2</a>, <a class="ref" href="bcialg_1.html#D19" onmouseover="rs('bcialg_1/D19')" onmouseout="rh()">BCIALG_1:def 19</a></span>;<br/>
<a NAME="E6:84_4"/><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="c13">x</font>,<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><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">RK</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#D12" onmouseover="rs('bcialg_2/D12')" onmouseout="rh()">BCIALG_2:def 12</a></span>;<br/>
<a NAME="E7:84_4"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Maroon" title="c13">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<a NAME="E8:84_4"/><span class="kw">then </span>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c6">RK1</font>,<font color="Maroon" title="c12">h</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="c6">RK1</font>,<span class="p1">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</a></span>;<br/>
<a NAME="E9:84_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E2:84_4"><span class="lab"><font color="Green" title="E49">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:84_4"/>
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_4"><span class="lab"><font color="Green" title="E49">A13</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><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_5"><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">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">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="c10">f</font> iff <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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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="c11">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="c10">f</font> iff <font color="Maroon" title="c11">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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span> )</span><br/><a NAME="E1:84_5"/>
( <font color="Maroon" title="c11">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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span> implies <font color="Maroon" title="c11">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="c10">f</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:84_5_1"/>
<font color="Maroon" title="c11">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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</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"> <font color="Maroon" title="c11">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="c10">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:84_5_1"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
<font color="Maroon" title="c12">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"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:84_5_1"/><span class="lab"><font color="Green" title="E50">A15</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c6">RK1</font>,<font color="Maroon" title="c12">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" onmouseover="rs('eqrel_1/D3')" onmouseout="rh()">EQREL_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E6:84_5_1"/><span class="lab"><font color="Green" title="E51">A16</font></span>: 
<font color="Maroon" title="c12">x</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">RK</font>,<font color="Maroon" title="c13">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_5_1"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="txt" href="bcialg_6.html#E70"><span class="lab"><font color="Green" title="E39">Lm5</font></span></a></span>;<br/>
<a NAME="E7:84_5_1"/>
( <font color="Maroon" title="c13">a</font> <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="c2">G</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="bcialg_1.html#D10" onmouseover="rs('bcialg_1/D10')" onmouseout="rh()">BCIALG_1:def 10</a></span>;<br/>
<a NAME="E8:84_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c6">RK1</font>,<font color="Maroon" title="c13">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E4:84_5_1"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="txt" href="#E6:84_5_1"><span class="lab"><font color="Green" title="E51">A16</font></span></a>, <a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</a></span>;<br/>
<a NAME="E9:84_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A17</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>;<br/>
<a NAME="E10:84_5_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">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="c2">G</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/>
<span class="kw">hence </span><a NAME="E11:84_5_1"/>
<font color="Maroon" title="c11">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="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:84_5_1"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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="E2:84_5"/>
( <font color="Maroon" title="c11">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="c10">f</font> iff <font color="Maroon" title="c11">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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</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="E11:84"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/>
<a NAME="E12:84"/><span class="kw">then </span>
<font color="Maroon" title="c10">f</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="E13:84"/>
<font color="Maroon" title="c2">G</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c8">RI</font>,<span class="p1">(<span class="default"><a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c6">RK1</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="#E9:84"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="bcialg_6.html#T54" target="_self" onmouseover="rs('bcialg_6/T54')" onmouseout="rh()">Th53</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>
