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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="freealg.html#V3" title="FREEALG:attr.3">free</a>  <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a><br/> for <font color="Olive" title="b3">h1</font>, <font color="Olive" title="b4">h2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b2">B</font>  st <font color="Olive" title="b3">h1</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h2</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b3">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b3">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">h2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a><br/> for <font color="Olive" title="b2">h1</font>, <font color="Olive" title="b3">h2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b2">h1</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b3">h2</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b2">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Olive" title="b2">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">h2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h1</font>, <font color="Olive" title="b2">h2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">h1</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b2">h2</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b1">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Olive" title="b1">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">h2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">h1</font>, <font color="Maroon" title="c5">h2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">h1</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">h2</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> implies <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">h2</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:60"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c4">h1</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E2:60"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c5">h2</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E3:60"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">h2</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> $1 holds <br/><font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>;<br/>
<a NAME="E4:60"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span><br/>

<a NAME="E2:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#T18" target="_self" title="AOFA_000:th.18">Th18</a></span>;<br/>
<a NAME="E3:60_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:60_1"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E2:60_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:60"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_2"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:60_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:60_2"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:60_2"/>
<font color="Maroon" title="c7">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E3:60_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">o</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c9">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <span class="kw"> such that </span><br/><a NAME="E6:60_2"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">and </span><br/><a NAME="E7:60_2"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:60_2"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="aofa_000.html#T20" target="_self" title="AOFA_000:th.20">Th20</a></span>;<br/>
<a NAME="E9:60_2"/>
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="alg_1.html#D1" title="ALG_1:def.1">ALG_1:def 1</a></span>;<br/>
<a NAME="E10:60_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T3" title="UNIALG_2:th.3">UNIALG_2:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">o9</font> = <font color="Maroon" title="c8">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">B</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E12:60_2"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
 <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font>
 
;<br/>
<a NAME="E13:60_2"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
 <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">B</font>
 
;<br/>
<a NAME="E14:60_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A15</font></span>: 
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c3">B</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="c4">h1</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c9">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E6:60_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E7:60_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E12:60_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="alg_1.html#D1" title="ALG_1:def.1">ALG_1:def 1</a></span>;<br/>
<a NAME="E15:60_2"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
<font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c10">o9</font>,<font color="Maroon" title="c3">B</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="c5">h2</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c9">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E6:60_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E7:60_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E12:60_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E13:60_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="alg_1.html#D1" title="ALG_1:def.1">ALG_1:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:60_2_1"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">h1</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">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E2:60_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">h2</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">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_2_1"/>
( <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">h1</font> &amp; <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">h2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2_1"><span class="lab"><font color="Green" title="E50">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font> holds <br/><font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font> implies <font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )</span><br/><span class="kw">assume </span><a NAME="E4:60_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span><br/><span class="kw">hence </span><a NAME="E5:60_2_1"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2"><span class="lab"><font color="Green" title="E40">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:60_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E8:60_2"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E14:60_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E15:60_2"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="aofa_000.html#T1" target="_self" title="AOFA_000:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:60"/><span class="lab"><font color="Green" title="E40">A18</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E5:60"><span class="lab"><font color="Green" title="E39">A6</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span><br/><a NAME="E1:60_3"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#T30" target="_self" title="AOFA_000:th.30">Th30</a></span>;<br/><span class="kw">hence </span><a NAME="E2:60_3"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60"><span class="lab"><font color="Green" title="E40">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:60"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
