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<span class="kw">let </span><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</font>, <font color="Maroon" title="c3">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">f</font> being  <a href="monoid_1.html#NM1" title="MONOID_1:NM.1">Function</a> of <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font>,<font color="Maroon" title="c3">D</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">A</font>,<font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b4">f2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">A</font>,<font color="Maroon" title="c2">D2</font><br/> for <font color="Olive" title="b5">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="monoid_1.html#K1" title="MONOID_1:func.1">..</a> <font color="Olive" title="b3">f1</font>,<font color="Olive" title="b4">f2</font>,<font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="monoid_1.html#NM1" title="MONOID_1:NM.1">Function</a> of <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font>,<font color="Maroon" title="c3">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">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b3">f2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">D2</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="monoid_1.html#K1" title="MONOID_1:func.1">..</a> <font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font>,<font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">A</font> be    <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">f1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b2">f2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">D2</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="monoid_1.html#K1" title="MONOID_1:func.1">..</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">f1</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">D1</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">f2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">D2</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="monoid_1.html#K1" title="MONOID_1:func.1">..</a> <font color="Maroon" title="c6">f1</font>,<font color="Olive" title="b1">f2</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">f2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">D2</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="monoid_1.html#K1" title="MONOID_1:func.1">..</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/>









<a NAME="E1:14"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f2</font> = <font color="Maroon" title="c7">f2</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">D2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E3:14"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c6">f1</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">D1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E5:14"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c8">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:14"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c8">f2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c8">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c8">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <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"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="monoid_1.html#K1" title="MONOID_1:func.1">..</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E7:14"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ; <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"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="monoid_1.html#K1" title="MONOID_1:func.1">..</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/>

<a NAME="E8:14"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">D2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c8">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="monoid_1.html#K1" title="MONOID_1:func.1">..</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="funct_5.html#T45" title="FUNCT_5:th.45">FUNCT_5:45</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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