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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b2">f</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b3">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b3">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Olive" title="b1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b1">f</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Olive" title="b1">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b2">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E2:48"/><span class="kw">then </span>
(  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun7c.html#T43" onmouseover="rs('mesfun7c/T43')" onmouseout="rh()">MESFUN7C:43</a></span>;<br/>
<a NAME="E3:48"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T72" onmouseover="rs('mesfunc6/T72')" onmouseout="rh()">MESFUNC6:72</a></span>;<br/>
<a NAME="E4:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A2</font></span>: 
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T5" onmouseover="rs('mesfun6c/T5')" onmouseout="rh()">MESFUN6C:5</a></span>;<br/>
<a NAME="E5:48"/>
(  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="mesfun7c.html#T43" onmouseover="rs('mesfun7c/T43')" onmouseout="rh()">MESFUN7C:43</a></span>;<br/>
<a NAME="E6:48"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T72" onmouseover="rs('mesfunc6/T72')" onmouseout="rh()">MESFUNC6:72</a></span>;<br/>
<a NAME="E7:48"/><span class="kw">then </span>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T5" onmouseover="rs('mesfun6c/T5')" onmouseout="rh()">MESFUN6C:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:48"/>
<font color="Maroon" title="c3">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">g</font> <a href="mesfun7c.html#R1" title="MESFUN7C:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="mesfun7c.html#T43" onmouseover="rs('mesfun7c/T43')" onmouseout="rh()">MESFUN7C:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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