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<span class="kw">let </span><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font>, <font color="Maroon" title="c3">f3</font>, <font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font>, <font color="Maroon" title="c6">g3</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<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">t0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Maroon" title="c2">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Maroon" title="c3">f3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Maroon" title="c6">g3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> holds <br/> <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">g2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g3</font></span>)</span>,<font color="Olive" title="b1">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">f3</font>,<font color="Olive" title="b1">t0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c6">g3</font>,<font color="Olive" title="b1">t0</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">t0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.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="c1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c2">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c3">f3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c6">g3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> implies  <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">g2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g3</font></span>)</span>,<font color="Maroon" title="c7">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:115"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
( <font color="Maroon" title="c1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c2">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c3">f3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> )
 <span class="kw">and </span><br/><a NAME="E2:115"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
( <font color="Maroon" title="c4">g1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> &amp; <font color="Maroon" title="c6">g3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">g2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g3</font></span>)</span>,<font color="Maroon" title="c7">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">p</font> = <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">q</font> = <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>;<br/>
<a NAME="E3:115"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
( <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font> &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font> &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E4:115"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
( <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font> &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font> &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
 <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">g2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g3</font></span>)</span>,<font color="Maroon" title="c7">t0</font> = 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">g2</font></span>)</span>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g3</font></span>)</span>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="fdiff_1.html#T21" title="FDIFF_1:th.21">FDIFF_1:21</a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g3</font></span>)</span>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="fdiff_1.html#T21" title="FDIFF_1:th.21">FDIFF_1:21</a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c7">t0</font></span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E3:115"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="fdiff_1.html#T21" title="FDIFF_1:th.21">FDIFF_1:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="euclid_8.html#E3"><span class="lab"><font color="Green" title="E3">Lm4</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:115"/>
 <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">g2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g3</font></span>)</span>,<font color="Maroon" title="c7">t0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">f3</font>,<font color="Maroon" title="c7">t0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c5">g2</font>,<font color="Maroon" title="c6">g3</font>,<font color="Maroon" title="c7">t0</font></span>)</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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