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<span class="kw">let </span><font color="Maroon" title="c1">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"> for <font color="Olive" title="b1">D</font> being 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> <br/> for <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">D</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Olive" title="b3">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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 <font color="Maroon" title="c2">D</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Olive" title="b2">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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 <font color="Maroon" title="c2">D</font>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Olive" title="b1">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c4">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:93"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E2:93"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
not <font color="Maroon" title="c4">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span><br/>

<a NAME="E3:93"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
( <font color="Maroon" title="c1">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="c4">f</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E4:93"/>
1 <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="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E5:93"/><span class="kw">then </span>
1 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>;<br/>
<a NAME="E6:93"/><span class="kw">then </span>
1 <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">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:93"/>
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T75" target="_self" title="FINSEQ_5:th.75">Th75</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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