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<span class="kw">let </span><font color="Maroon" title="c1">r1</font>, <font color="Maroon" title="c2">r2</font>, <font color="Maroon" title="c3">x0</font>, <font color="Maroon" title="c4">x1</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"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b2">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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"> <span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></span>)</span></span><br/>











<span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span> = 
<span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f1</font></span>)</span>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T32" target="_self" title="DIFF_1:th.32">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T31" target="_self" title="DIFF_1:th.31">Th31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="diff_1.html#T31" target="_self" title="DIFF_1:th.31">Th31</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:50"/>
<span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c4">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span></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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