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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">F</font> being   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> holds <br/> (  <a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Olive" title="b1">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  iff  <a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Olive" title="b1">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</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"> (  <a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  iff  <a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  )</span><br/>



<div><span class="kw">hereby </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"> (  <a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  implies  <a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:28_1"/>
 <a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</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">  <a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> </span><br/><a NAME="E2:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#T24" target="_self" onmouseover="rs('mesfun9c/T24')" onmouseout="rh()">Th24</a></span>;<br/><a NAME="E3:28_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun7c.html#D12" onmouseover="rs('mesfun7c/D12')" onmouseout="rh()">MESFUN7C:def 12</a>, <a class="ref" href="mesfunc8.html#D2" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">MESFUNC8:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:28_1"/>
 <a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</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"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:28"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
 <a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</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">  <a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">now </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">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font></span>)</span></span><br/><a NAME="E1:28_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun7c.html#D12" onmouseover="rs('mesfun7c/D12')" onmouseout="rh()">MESFUN7C:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E2:28_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c4">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E29">A1</font></span></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/></div><span class="kw">end;</span></div>
<a NAME="E4:28"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E5:28"/>
 <a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</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"> verum</span><br/>


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