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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <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">R</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b1">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Olive" title="b3">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">d</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <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">R</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Olive" title="b2">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Olive" title="b1">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">R</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:53"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c3">f</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="c2">d</font> <a href="rfunct_3.html#R1" title="RFUNCT_3:pred.1">is_common_for_dom</a> <font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">m</font> =  <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">G</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rfunct_3.html#D9" target="_self" title="RFUNCT_3:def.9">RFUNCT_3:def 9</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">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> holds <br/><font color="Maroon" title="c2">d</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="c6">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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"><font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> implies <font color="Maroon" title="c2">d</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="c6">G</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c7">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"><font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:53"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c2">d</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="c6">G</font></span><br/>

<a NAME="E4:53"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D7" target="_self" title="RFUNCT_3:def.7">Def7</a></span>;<br/>
<a NAME="E5:53"/><span class="kw">then </span>
(  <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<a NAME="E6:53"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:53"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:53"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c7">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> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/>
<a NAME="E10:53"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c2">d</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">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="rfunct_3.html#D9" target="_self" title="RFUNCT_3:def.9">Def9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c4">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E12:53"/>
<font color="Maroon" title="c6">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">r</font> <a href="rfunct_3.html#K13" title="RFUNCT_3:func.13">(#)</a> <font color="Maroon" title="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="rfunct_3.html#D7" target="_self" title="RFUNCT_3:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:53"/>
<font color="Maroon" title="c2">d</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="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:53"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</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/>


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