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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">H</font> being   <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Olive" title="b2">H</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Olive" title="b2">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b3">x</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b3">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being   <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Olive" title="b1">H</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Olive" title="b1">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b2">x</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b2">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">H</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c3">H</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c3">H</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E32">A1</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c3">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span><br/><a NAME="E1:47_1"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E2:47_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:47"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self" title="SEQFUNC:def.10">Def10</a></i>;<br/><a NAME="E3:47_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:47_1"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/><a NAME="E4:47_1"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D2" target="_self" title="SEQFUNC:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E4:47_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="seqfunc.html#K2" title="SEQFUNC:func.2">(#)</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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