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<span class="kw">let </span><font color="Maroon" title="c1">U1</font> be   <a href="unialg_1.html#NM1" title="UNIALG_1:NM.1">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="alg_1.html#M1" title="ALG_1:mode.1">Congruence</a> of <font color="Maroon" title="c1">U1</font> holds   <a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Olive" title="b1">E</font>) <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>, <a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Olive" title="b1">E</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be    <a href="alg_1.html#M1" title="ALG_1:mode.1">Congruence</a> of <font color="Maroon" title="c1">U1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>) <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>, <a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">f</font> =  <a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">u1</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">qu</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>;<br/>
<a NAME="E1:27"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <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="c1">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_1.html#D4" title="UNIALG_1:def.4">UNIALG_1:def 4</a></span>;<br/>
<a NAME="E2:27"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:27"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" title="ALG_1:func.3">QuotOpSeq</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <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="c1">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#D9" target="_self" title="ALG_1:def.9">Def13</a></span>;<br/>
<a NAME="E4:27"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_1.html#D4" title="UNIALG_1:def.4">UNIALG_1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> implies <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span>
 ; <a class="txt" 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="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><a NAME="E2:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">U1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">o1</font> =  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> as   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:27_1"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" title="ALG_1:func.3">QuotOpSeq</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<span class="p1">(<span class="default"><a href="alg_1.html#K3" title="ALG_1:func.3">QuotOpSeq</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="alg_1.html#K2" title="ALG_1:func.2">QuotOp</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#D9" target="_self" title="ALG_1:def.9">Def13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">cl</font> =  <a href="alg_1.html#K2" title="ALG_1:func.2">QuotOp</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font>) as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="margrel1.html#V2" title="MARGREL1:attr.2">homogeneous</a>   <a href="margrel1.html#V3" title="MARGREL1:attr.3">quasi_total</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c9">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:27_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">cl</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">b</font> as    <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 <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a></span>;<br/><a NAME="E10:27_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K2" title="ALG_1:func.2">QuotOp</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#D8" target="_self" title="ALG_1:def.8">Def12</a></span>;<br/><a NAME="E11:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></span>;<br/><a NAME="E12:27_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font> )
 ;<br/><a NAME="E13:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
 <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c8">cl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="margrel1.html#D25" title="MARGREL1:def.25">MARGREL1:def 25</a></span>;<br/><a NAME="E14:27_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="unialg_1.html#D4" title="UNIALG_1:def.4">UNIALG_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E15:27_1"/>
<span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E13:27_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="unialg_1.html#D4" title="UNIALG_1:def.4">UNIALG_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:27"/>
 <a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K1" title="UNIALG_1:func.1">signature</a> <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="unialg_2.html#D1" title="UNIALG_2:def.1">UNIALG_2:def 1</a>,<a class="ref" href="alg_1.html#D1" target="_self" title="ALG_1:def.1">ALG_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">U1</font> holds <br/> for <font color="Olive" title="b2">o1</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U1</font><br/>  for <font color="Olive" title="b3">o2</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>  st <font color="Olive" title="b2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/> for <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b4">x</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">U1</font> implies  for <font color="Olive" title="b1">o1</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U1</font><br/>  for <font color="Olive" title="b2">o2</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>  st <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Olive" title="b2">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b3">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E7:27"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">U1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o1</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U1</font><br/>  for <font color="Olive" title="b2">o2</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>  st <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Olive" title="b2">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b3">x</font></span>)</span></span><br/>

<a NAME="E8:27"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><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">U1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K3" title="ALG_1:func.3">QuotOpSeq</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">o1</font> be   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o2</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>  st <font color="Maroon" title="c7">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Olive" title="b1">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</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="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">o2</font> be   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> implies  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</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="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E10:27"/>
(  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o1</font> &amp; <font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K4" title="ALG_1:func.4">QuotUnivAlg</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</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="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/>

<a NAME="E11:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="alg_1.html#K2" title="ALG_1:func.2">QuotOp</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="alg_1.html#D9" target="_self" title="ALG_1:def.9">Def13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">o1</font> implies <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</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="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c9">x</font> as    <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">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">fx</font> = <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">fx</font> = <font color="Maroon" title="c11">fx</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E15:27"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">x</font> implies  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:27_2"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span><br/><a NAME="E2:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></span>;<br/><a NAME="E3:27_2"/>
<font color="Maroon" title="c9">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">xm</font> = <font color="Maroon" title="c9">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> ;<br/><span class="kw">thus </span> <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>) = 
<span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">xm</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#D11" target="_self" title="ALG_1:def.11">Def15</a></span>
<br/>.= 
<font color="Maroon" title="c12">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_2"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></span>
; <a class="txt" 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="E17:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="finseq_3.html#R1" title="FINSEQ_3:pred.1">is_representatives_FS</a> <font color="Maroon" title="c12">fx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="finseq_3.html#D4" title="FINSEQ_3:def.4">FINSEQ_3:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E18:27"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">o1</font>
 ; <a class="txt" 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="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</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="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span><br/>

<a NAME="E19:27"/><span class="kw">then </span>
<font color="Maroon" title="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">ox</font> = <font color="Maroon" title="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> ;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">o1</font> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T22" title="MARGREL1:th.22">MARGREL1:22</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <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">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font> </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></span>
;<br/>
<a NAME="E22:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A16</font></span>: 
 ex <font color="Olive" title="b1">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">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:27"><span class="lab"><font color="Green" title="E30">A15</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E32">A17</font></span>: <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</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="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> = 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c13">ox</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#D11" target="_self" title="ALG_1:def.11">Def15</a></span>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>)

;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K2" title="ALG_1:func.2">QuotOp</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#D8" target="_self" title="ALG_1:def.8">Def12</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o1</font> </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></span>
;<br/>
<a NAME="E25:27"/><span class="kw">then </span>
<font color="Maroon" title="c12">fx</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="alg_1.html#K2" title="ALG_1:func.2">QuotOp</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:27"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:27"/>
<span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</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="c7">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">Nat_Hom</a> (<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:27"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E11:27"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="alg_1.html#D8" target="_self" title="ALG_1:def.8">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
