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<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">f</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font> holds   <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Olive" title="b1">f</font> is   <a href="vectsp_8.html#NM4" title="VECTSP_8:NM.4">sup-Semilattice-Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</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">  <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font> is   <a href="vectsp_8.html#NM4" title="VECTSP_8:NM.4">sup-Semilattice-Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font></span>)</span></span><br/>



<a NAME="E1:32"/>
 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 <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <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="lattices.html#K3" title="LATTICES:func.3">"\/"</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"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</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"> <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">A1</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latsubgr.html#T2" target="_self" onmouseover="rs('latsubgr/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">B1</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E4:32_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c7">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latsubgr.html#T2" target="_self" onmouseover="rs('latsubgr/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:32_1"/>
<span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> holds <br/><font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</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="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> implies <font color="Maroon" title="c8">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</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="c8">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<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="c7">B1</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E5:32_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E6:32_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" onmouseover="rs('group_2/T51')" onmouseout="rh()">GROUP_2:51</a></span>;<br/>
<a NAME="E7:32_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="group_1.html#K2" title="GROUP_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="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E8:32_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T32" onmouseover="rs('group_6/T32')" onmouseout="rh()">GROUP_6:32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:32_1_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E7:32_1_1_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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="E2:32_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" onmouseover="rs('group_2/T46')" onmouseout="rh()">GROUP_2:46</a></span>;<br/>
<a NAME="E3:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A7</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</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="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E4:32_1_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font> st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 
;<br/>
<a NAME="E5:32_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</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">G1</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</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">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" onmouseover="rs('group_2/D5')" onmouseout="rh()">GROUP_2:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">A</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E7:32_1_1"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> holds <br/><font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</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">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> implies <font color="Maroon" title="c9">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_2"/>
<font color="Maroon" title="c9">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</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">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_2"/><span class="lab"><font color="Green" title="E23">A10</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="c6">A1</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_2"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E5:32_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_2"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E6:32_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" onmouseover="rs('group_2/T51')" onmouseout="rh()">GROUP_2:51</a></span>;<br/>
<a NAME="E7:32_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="group_1.html#K2" title="GROUP_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="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E8:32_1_1_2"/>
<font color="Maroon" title="c3">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">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T32" onmouseover="rs('group_6/T32')" onmouseout="rh()">GROUP_6:32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:32_1_1_2"/>
<font color="Maroon" title="c9">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E7:32_1_1_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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">reconsider </span><font color="Maroon" title="c9">B</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G2</font> ;<br/>
<a NAME="E9:32_1_1"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</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="E10:32_1_1"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> &amp; <font color="Olive" title="b2">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">g1</font>, <font color="Maroon" title="c11">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</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="c10">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> &amp; <font color="Maroon" title="c11">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> implies <font color="Maroon" title="c10">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_1_1_3"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c10">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font>
 <span class="kw">and </span><br/><a NAME="E2:32_1_1_3"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c11">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</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="c10">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_3"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<font color="Maroon" title="c12">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="c7">B1</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_3"/><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c10">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_3"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E7:32_1_1_3"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<font color="Maroon" title="c13">y</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="c7">B1</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1_1_3"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
<font color="Maroon" title="c11">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_3"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E9:32_1_1_3"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_3"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E10:32_1_1_3"/>
<font color="Maroon" title="c12">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_3"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E11:32_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1_3"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<a NAME="E12:32_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A22</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">y</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="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:32_1_1_3"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:32_1_1_3"/>
<font color="Maroon" title="c10">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_3"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E8:32_1_1_3"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E12:32_1_1_3"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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="E11:32_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" onmouseover="rs('group_2/T46')" onmouseout="rh()">GROUP_2:46</a></span>;<br/>
<a NAME="E12:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A23</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</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="c7">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:32_1_1"/><span class="lab"><font color="Green" title="E26">A24</font></span>: 
<span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">B1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latsubgr.html#D3" target="_self" onmouseover="rs('latsubgr/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E14:32_1_1"/>
 ex <font color="Olive" title="b1">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font> st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 
;<br/>
<a NAME="E15:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E12:32_1_1"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <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">then </span><span class="kw">consider </span><font color="Maroon" title="c10">B3</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E17:32_1_1"/><span class="lab"><font color="Green" title="E27">A25</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="group_2.html#T52" onmouseover="rs('group_2/T52')" onmouseout="rh()">GROUP_2:52</a></span>;<br/>
<a NAME="E18:32_1_1"/><span class="lab"><font color="Green" title="E28">A26</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> &amp; <font color="Olive" title="b2">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</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="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> implies <font color="Maroon" title="c11">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_1_1_4"/><span class="lab"><font color="Green" title="E29">A27</font></span>: 
<font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font>
 <span class="kw">and </span><br/><a NAME="E2:32_1_1_4"/><span class="lab"><font color="Green" title="E30">A28</font></span>: 
<font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</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="c11">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font></span><br/>

<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="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_4"/><span class="lab"><font color="Green" title="E31">A29</font></span>: 
<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="c6">A1</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_4"/><span class="lab"><font color="Green" title="E32">A30</font></span>: 
<font color="Maroon" title="c11">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_4"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E7:32_1_1_4"/><span class="lab"><font color="Green" title="E33">A31</font></span>: 
<font color="Maroon" title="c14">y</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="c6">A1</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1_1_4"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c12">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_4"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E9:32_1_1_4"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_4"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E10:32_1_1_4"/>
<font color="Maroon" title="c13">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_4"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E11:32_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1_4"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<a NAME="E12:32_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A34</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</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="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:32_1_1_4"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:32_1_1_4"/>
<font color="Maroon" title="c11">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_4"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E8:32_1_1_4"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="txt" href="#E12:32_1_1_4"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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="E19:32_1_1"/><span class="lab"><font color="Green" title="E29">A35</font></span>: 
( <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="latsubgr.html#D3" target="_self" onmouseover="rs('latsubgr/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">C1</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E21:32_1_1"/><span class="lab"><font color="Green" title="E30">A36</font></span>: 
<font color="Maroon" title="c11">C1</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">A1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">B1</font>
 ;<br/>
<a NAME="E22:32_1_1"/><span class="lab"><font color="Green" title="E31">A37</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> &amp; <font color="Olive" title="b2">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">g1</font>, <font color="Maroon" title="c13">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</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">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> &amp; <font color="Maroon" title="c13">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> implies <font color="Maroon" title="c12">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_1_1_5"/><span class="lab"><font color="Green" title="E32">A38</font></span>: 
<font color="Maroon" title="c12">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font>
 <span class="kw">and </span><br/><a NAME="E2:32_1_1_5"/><span class="lab"><font color="Green" title="E33">A39</font></span>: 
<font color="Maroon" title="c13">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</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">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c14">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_5"/><span class="lab"><font color="Green" title="E34">A40</font></span>: 
<font color="Maroon" title="c14">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="c11">C1</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_5"/><span class="lab"><font color="Green" title="E35">A41</font></span>: 
<font color="Maroon" title="c12">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_5"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E7:32_1_1_5"/><span class="lab"><font color="Green" title="E36">A42</font></span>: 
<font color="Maroon" title="c15">y</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="c11">C1</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1_1_5"/><span class="lab"><font color="Green" title="E37">A43</font></span>: 
<font color="Maroon" title="c13">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_5"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E9:32_1_1_5"/><span class="lab"><font color="Green" title="E38">A44</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_5"><span class="lab"><font color="Green" title="E36">A42</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E10:32_1_1_5"/>
<font color="Maroon" title="c14">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_5"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E11:32_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1_5"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<a NAME="E12:32_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A45</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">y</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="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:32_1_1_5"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:32_1_1_5"/>
<font color="Maroon" title="c12">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_5"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="txt" href="#E8:32_1_1_5"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="txt" href="#E12:32_1_1_5"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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="E23:32_1_1"/><span class="lab"><font color="Green" title="E32">A46</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> holds <br/><font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</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">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> implies <font color="Maroon" title="c12">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_6"/>
<font color="Maroon" title="c12">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</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">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</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="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_6"/><span class="lab"><font color="Green" title="E33">A47</font></span>: 
<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="c11">C1</font>
 <span class="kw">and </span><br/><a NAME="E4:32_1_1_6"/><span class="lab"><font color="Green" title="E34">A48</font></span>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E5:32_1_1_6"/>
<font color="Maroon" title="c13">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_6"><span class="lab"><font color="Green" title="E33">A47</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E6:32_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" onmouseover="rs('group_2/T51')" onmouseout="rh()">GROUP_2:51</a></span>;<br/>
<a NAME="E7:32_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A49</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="group_1.html#K2" title="GROUP_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="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E8:32_1_1_6"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T32" onmouseover="rs('group_6/T32')" onmouseout="rh()">GROUP_6:32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:32_1_1_6"/>
<font color="Maroon" title="c12">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_6"><span class="lab"><font color="Green" title="E34">A48</font></span></a>, <a class="txt" href="#E7:32_1_1_6"><span class="lab"><font color="Green" title="E35">A49</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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="E24:32_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" onmouseover="rs('group_2/T46')" onmouseout="rh()">GROUP_2:46</a></span>;<br/>
<a NAME="E25:32_1_1"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</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="c11">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E26:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <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">then </span><span class="kw">consider </span><font color="Maroon" title="c12">C3</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E28:32_1_1"/><span class="lab"><font color="Green" title="E33">A50</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">C3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:32_1_1"><span class="lab"><font color="Green" title="E32">A46</font></span></a>, <a class="txt" href="#E22:32_1_1"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="ref" href="group_2.html#T52" onmouseover="rs('group_2/T52')" onmouseout="rh()">GROUP_2:52</a></span>;<br/>
<a NAME="E29:32_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font> st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 
;<br/>
<a NAME="E30:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <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">then </span><span class="kw">consider </span><font color="Maroon" title="c13">A3</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E32:32_1_1"/><span class="lab"><font color="Green" title="E34">A51</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E18:32_1_1"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="ref" href="group_2.html#T52" onmouseover="rs('group_2/T52')" onmouseout="rh()">GROUP_2:52</a></span>;<br/>
<a NAME="E33:32_1_1"/><span class="lab"><font color="Green" title="E35">A52</font></span>: 
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font></span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_1_1"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="latsubgr.html#T3" target="_self" onmouseover="rs('latsubgr/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E34:32_1_1"/>
 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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</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="c12">C3</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_7"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_1_7"/><span class="lab"><font color="Green" title="E36">A53</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</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="c12">C3</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> or <font color="Maroon" title="c14">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 <font color="Maroon" title="c12">C3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_7_1"/><span class="lab"><font color="Green" title="E37">A54</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</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="c14">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 <font color="Maroon" title="c12">C3</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_7_1"/><span class="lab"><font color="Green" title="E38">A55</font></span>: 
<font color="Maroon" title="c16">y</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="c7">B1</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_7_1"/><span class="lab"><font color="Green" title="E39">A56</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7_1"><span class="lab"><font color="Green" title="E37">A54</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E6:32_1_1_7_1"/>
<font color="Maroon" title="c16">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_7_1"><span class="lab"><font color="Green" title="E38">A55</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E7:32_1_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T29" onmouseover="rs('group_4/T29')" onmouseout="rh()">GROUP_4:29</a></span>;<br/>
<a NAME="E8:32_1_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</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"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c8">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E9:32_1_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</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="c6">A1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">B1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latsubgr.html#T4" target="_self" onmouseover="rs('latsubgr/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:32_1_1_7_1"/>
<font color="Maroon" title="c14">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 <font color="Maroon" title="c12">C3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="txt" href="#E28:32_1_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E5:32_1_1_7_1"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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="E2:32_1_1_7"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</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="c12">C3</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> or <font color="Maroon" title="c14">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 <font color="Maroon" title="c12">C3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_7_2"/><span class="lab"><font color="Green" title="E37">A57</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</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="c14">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 <font color="Maroon" title="c12">C3</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_7_2"/><span class="lab"><font color="Green" title="E38">A58</font></span>: 
<font color="Maroon" title="c16">y</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="c6">A1</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_7_2"/><span class="lab"><font color="Green" title="E39">A59</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7_2"><span class="lab"><font color="Green" title="E37">A57</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E6:32_1_1_7_2"/>
<font color="Maroon" title="c16">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_7_2"><span class="lab"><font color="Green" title="E38">A58</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E7:32_1_1_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T29" onmouseover="rs('group_4/T29')" onmouseout="rh()">GROUP_4:29</a></span>;<br/>
<a NAME="E8:32_1_1_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</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"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c8">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E9:32_1_1_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</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="c6">A1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">B1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latsubgr.html#T4" target="_self" onmouseover="rs('latsubgr/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:32_1_1_7_2"/>
<font color="Maroon" title="c14">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 <font color="Maroon" title="c12">C3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="txt" href="#E28:32_1_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E5:32_1_1_7_2"><span class="lab"><font color="Green" title="E39">A59</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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="E3:32_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c9">B</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="c12">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7"><span class="lab"><font color="Green" title="E36">A53</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E4:32_1_1_7"/><span class="kw">then </span>
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c9">B</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c12">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#D4" onmouseover="rs('group_4/D4')" onmouseout="rh()">GROUP_4:def 4</a></span>;<br/>
<a NAME="E5:32_1_1_7"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c9">B</font></span>)</span> <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="c12">C3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" onmouseover="rs('group_2/D5')" onmouseout="rh()">GROUP_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:32_1_1_7"/>
 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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> <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="c12">C3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:32_1_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E17:32_1_1"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="latsubgr.html#T4" target="_self" onmouseover="rs('latsubgr/T4')" onmouseout="rh()">Th4</a></span>; <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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">C3</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"><font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">AA</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">A3</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B3</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G1</font> ;<br/>
<a NAME="E8:32_1_1_7"/><span class="lab"><font color="Green" title="E37">A60</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font>  st <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_7_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</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="c15">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> implies <font color="Maroon" title="c15">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_7_3"/>
<font color="Maroon" title="c15">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</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="c15">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span><br/>

<a NAME="E2:32_1_1_7_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">g</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</a></span>;<br/>
<a NAME="E3:32_1_1_7_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E4:32_1_1_7_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" onmouseover="rs('group_2/T51')" onmouseout="rh()">GROUP_2:51</a></span>;<br/>
<a NAME="E5:32_1_1_7_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T32" onmouseover="rs('group_6/T32')" onmouseout="rh()">GROUP_6:32</a></span>;<br/>
<a NAME="E6:32_1_1_7_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:32_1_1_7_3"/>
<font color="Maroon" title="c15">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</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="E9:32_1_1_7"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</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">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" onmouseover="rs('group_2/D5')" onmouseout="rh()">GROUP_2:def 5</a></span>;<br/>
<a NAME="E10:32_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A61</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_1_1"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="funct_2.html#T42" onmouseover="rs('funct_2/T42')" onmouseout="rh()">FUNCT_2:42</a></span>;<br/>
<a NAME="E11:32_1_1_7"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</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">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" onmouseover="rs('group_2/D5')" onmouseout="rh()">GROUP_2:def 5</a></span>;<br/>
<a NAME="E12:32_1_1_7"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:32_1_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="ref" href="funct_2.html#T42" onmouseover="rs('funct_2/T42')" onmouseout="rh()">FUNCT_2:42</a></span>;<br/>
<a NAME="E13:32_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A62</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">AA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1_7"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="xboole_1.html#T13" onmouseover="rs('xboole_1/T13')" onmouseout="rh()">XBOOLE_1:13</a></span>;<br/>
<a NAME="E14:32_1_1_7"/><span class="lab"><font color="Green" title="E40">A63</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font>  st <font color="Olive" title="b1">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> &amp; <font color="Olive" title="b2">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> holds <br/><font color="Olive" title="b1">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_7_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">g1</font>, <font color="Maroon" title="c16">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</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="c15">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> &amp; <font color="Maroon" title="c16">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> implies <font color="Maroon" title="c15">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_1_1_7_4"/><span class="lab"><font color="Green" title="E41">A64</font></span>: 
<font color="Maroon" title="c15">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:32_1_1_7_4"/><span class="lab"><font color="Green" title="E42">A65</font></span>: 
<font color="Maroon" title="c16">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</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="c15">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span><br/>

<a NAME="E3:32_1_1_7_4"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g2</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_7_4"><span class="lab"><font color="Green" title="E42">A65</font></span></a>, <a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</a></span>;<br/>
<a NAME="E4:32_1_1_7_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A66</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E5:32_1_1_7_4"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">g1</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7_4"><span class="lab"><font color="Green" title="E41">A64</font></span></a>, <a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</a></span>;<br/>
<a NAME="E6:32_1_1_7_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E7:32_1_1_7_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">g1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">g2</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_7_4"><span class="lab"><font color="Green" title="E43">A66</font></span></a>, <a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<a NAME="E8:32_1_1_7_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g2</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>;<br/>
<a NAME="E9:32_1_1_7_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g2</font></span>)</span> <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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:32_1_1_7_4"/>
<font color="Maroon" title="c15">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</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="E15:32_1_1_7"/><span class="lab"><font color="Green" title="E41">A67</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_7_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</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="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B3</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_7_5"/><span class="lab"><font color="Green" title="E42">A68</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B3</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="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span><br/>

<a NAME="E2:32_1_1_7_5"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</a></span>;<br/>
<a NAME="E3:32_1_1_7_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A69</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E4:32_1_1_7_5"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7_5"><span class="lab"><font color="Green" title="E42">A68</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<a NAME="E5:32_1_1_7_5"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_7_5"><span class="lab"><font color="Green" title="E43">A69</font></span></a>, <a class="ref" href="latsubgr.html#T5" target="_self" onmouseover="rs('latsubgr/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E6:32_1_1_7_5"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:32_1_1_7_5"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7_5"><span class="lab"><font color="Green" title="E42">A68</font></span></a>, <a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</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="E16:32_1_1_7"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" onmouseover="rs('group_2/T46')" onmouseout="rh()">GROUP_2:46</a></span>;<br/>
<a NAME="E17:32_1_1_7"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G2</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E18:32_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span> <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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T31" onmouseover="rs('group_6/T31')" onmouseout="rh()">GROUP_6:31</a></span>;<br/>
<a NAME="E19:32_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">H</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E21:32_1_1_7"/><span class="lab"><font color="Green" title="E42">A70</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1_7"><span class="lab"><font color="Green" title="E37">A60</font></span></a>, <a class="txt" href="#E14:32_1_1_7"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="ref" href="group_2.html#T52" onmouseover="rs('group_2/T52')" onmouseout="rh()">GROUP_2:52</a></span>;<br/>
<a NAME="E22:32_1_1_7"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_7_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">A3</font> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_7_6"/><span class="lab"><font color="Green" title="E43">A71</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">A3</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span><br/>

<a NAME="E2:32_1_1_7_6"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</a></span>;<br/>
<a NAME="E3:32_1_1_7_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A72</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E4:32_1_1_7_6"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7_6"><span class="lab"><font color="Green" title="E43">A71</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<a NAME="E5:32_1_1_7_6"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_7_6"><span class="lab"><font color="Green" title="E44">A72</font></span></a>, <a class="ref" href="latsubgr.html#T5" target="_self" onmouseover="rs('latsubgr/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E6:32_1_1_7_6"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:32_1_1_7_6"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7_6"><span class="lab"><font color="Green" title="E43">A71</font></span></a>, <a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</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="E23:32_1_1_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">A3</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">B3</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1_7"><span class="lab"><font color="Green" title="E41">A67</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E24:32_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c8">A</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="c15">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:32_1_1_7"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="txt" href="#E21:32_1_1_7"><span class="lab"><font color="Green" title="E42">A70</font></span></a></span>;<br/>
<a NAME="E25:32_1_1_7"/><span class="kw">then </span>
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c8">A</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c15">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#D4" onmouseover="rs('group_4/D4')" onmouseout="rh()">GROUP_4:def 4</a></span>;<br/>
<a NAME="E26:32_1_1_7"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c8">A</font></span>)</span> <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="c15">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" onmouseover="rs('group_2/D5')" onmouseout="rh()">GROUP_2:def 5</a></span>;<br/>
<a NAME="E27:32_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A73</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="txt" href="#E21:32_1_1_7"><span class="lab"><font color="Green" title="E42">A70</font></span></a>, <a class="ref" href="latsubgr.html#T4" target="_self" onmouseover="rs('latsubgr/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E28:32_1_1_7"/><span class="lab"><font color="Green" title="E44">A74</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_7_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</font> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_7_7"/><span class="lab"><font color="Green" title="E45">A75</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C1</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c16">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font> ;<br/>
<a NAME="E3:32_1_1_7_7"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font> st <br/>( <font color="Olive" title="b1">y</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="c11">C1</font> &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_7_7"><span class="lab"><font color="Green" title="E45">A75</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:32_1_1_7_7"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:32_1_1_7"><span class="lab"><font color="Green" title="E43">A73</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</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="E29:32_1_1_7"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T75" onmouseover="rs('funct_1/T75')" onmouseout="rh()">FUNCT_1:75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:32_1_1_7"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">C3</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"><font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:32_1_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="txt" href="#E28:32_1_1_7"><span class="lab"><font color="Green" title="E44">A74</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> <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c7">B1</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c13">A3</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c10">B3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="txt" href="#E28:32_1_1"><span class="lab"><font color="Green" title="E33">A50</font></span></a>, <a class="ref" href="latsubgr.html#T3" target="_self" onmouseover="rs('latsubgr/T3')" onmouseout="rh()">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">A1</font></span>)</span></span>)</span> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">B1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:32_1_1"><span class="lab"><font color="Green" title="E34">A51</font></span></a>, <a class="txt" href="#E33:32_1_1"><span class="lab"><font color="Green" title="E35">A52</font></span></a>, <a class="ref" href="latsubgr.html#T3" target="_self" onmouseover="rs('latsubgr/T3')" onmouseout="rh()">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:32_1_1"><span class="lab"><font color="Green" title="E29">A35</font></span></a>, <a class="ref" href="latsubgr.html#T22" target="_self" onmouseover="rs('latsubgr/T22')" onmouseout="rh()">Th22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E36:32_1_1"/>
<span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E13:32_1_1"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="ref" href="latsubgr.html#T22" target="_self" onmouseover="rs('latsubgr/T22')" onmouseout="rh()">Th22</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:32"/>
 <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font> is   <a href="vectsp_8.html#NM4" title="VECTSP_8:NM.4">sup-Semilattice-Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c2">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D1" onmouseover="rs('lattice4/D1')" onmouseout="rh()">LATTICE4:def 1</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>
