let f1, f2 be PartFunc of COMPLEX,COMPLEX; :: thesis: for x0 being Complex st f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds
( f1 + f2 is_differentiable_in x0 & diff ((f1 + f2),x0) = (diff (f1,x0)) + (diff (f2,x0)) )
let x0 be Complex; :: thesis: ( f1 is_differentiable_in x0 & f2 is_differentiable_in x0 implies ( f1 + f2 is_differentiable_in x0 & diff ((f1 + f2),x0) = (diff (f1,x0)) + (diff (f2,x0)) ) )
assume that
A1: f1 is_differentiable_in x0 and
A2: f2 is_differentiable_in x0 ; :: thesis: ( f1 + f2 is_differentiable_in x0 & diff ((f1 + f2),x0) = (diff (f1,x0)) + (diff (f2,x0)) )
consider N1 being Neighbourhood of x0 such that
A3: N1 c= dom f1 and
A4: ex L being C_LinearFunc ex R being C_RestFunc st
for x being Complex st x in N1 holds
(f1 /. x) - (f1 /. x0) = (L /. (x - x0)) + (R /. (x - x0)) by A1;
consider L1 being C_LinearFunc, R1 being C_RestFunc such that
A5: for x being Complex st x in N1 holds
(f1 /. x) - (f1 /. x0) = (L1 /. (x - x0)) + (R1 /. (x - x0)) by A4;
consider N2 being Neighbourhood of x0 such that
A6: N2 c= dom f2 and
A7: ex L being C_LinearFunc ex R being C_RestFunc st
for x being Complex st x in N2 holds
(f2 /. x) - (f2 /. x0) = (L /. (x - x0)) + (R /. (x - x0)) by A2;
consider L2 being C_LinearFunc, R2 being C_RestFunc such that
A8: for x being Complex st x in N2 holds
(f2 /. x) - (f2 /. x0) = (L2 /. (x - x0)) + (R2 /. (x - x0)) by A7;
reconsider R = R1 + R2 as C_RestFunc ;
reconsider L = L1 + L2 as C_LinearFunc ;
consider N being Neighbourhood of x0 such that
A9: N c= N1 and
A10: N c= N2 by Lm1;
A11: N c= dom f2 by A6, A10;
N c= dom f1 by A3, A9;
then N /\ N c= (dom f1) /\ (dom f2) by A11, XBOOLE_1:27;
then A12: N c= dom (f1 + f2) by VALUED_1:def 1;

A13: now :: thesis: for x being Complex st x in N holds
((f1 + f2) /. x) - ((f1 + f2) /. x0) = (L /. (x - x0)) + (R /. (x - x0))

let x be Complex; :: thesis: ( x in N implies ((f1 + f2) /. x) - ((f1 + f2) /. x0) = (L /. (x - x0)) + (R /. (x - x0)) )
A14: x0 in N by Th7;
A15: x - x0 in COMPLEX by XCMPLX_0:def 2;
assume A16: x in N ; :: thesis: ((f1 + f2) /. x) - ((f1 + f2) /. x0) = (L /. (x - x0)) + (R /. (x - x0))
hence ((f1 + f2) /. x) - ((f1 + f2) /. x0) = ((f1 /. x) + (f2 /. x)) - ((f1 + f2) /. x0) by A12, CFUNCT_1:1
.= ((f1 /. x) + (f2 /. x)) - ((f1 /. x0) + (f2 /. x0)) by A12, A14, CFUNCT_1:1
.= ((f1 /. x) - (f1 /. x0)) + ((f2 /. x) - (f2 /. x0))
.= ((L1 /. (x - x0)) + (R1 /. (x - x0))) + ((f2 /. x) - (f2 /. x0)) by A5, A9, A16
.= ((L1 /. (x - x0)) + (R1 /. (x - x0))) + ((L2 /. (x - x0)) + (R2 /. (x - x0))) by A8, A10, A16
.= ((L1 /. (x - x0)) + (L2 /. (x - x0))) + ((R1 /. (x - x0)) + (R2 /. (x - x0)))
.= (L /. (x - x0)) + ((R1 /. (x - x0)) + (R2 /. (x - x0))) by CFUNCT_1:64, A15
.= (L /. (x - x0)) + (R /. (x - x0)) by CFUNCT_1:64, A15 ;
:: thesis: verum

end;

hence f1 + f2 is_differentiable_in x0 by A12; :: thesis: diff ((f1 + f2),x0) = (diff (f1,x0)) + (diff (f2,x0))
hence diff ((f1 + f2),x0) = L /. 1r by A12, A13, Def7
.= (L1 /. 1r) + (L2 /. 1r) by CFUNCT_1:64
.= (diff (f1,x0)) + (L2 /. 1r) by A1, A3, A5, Def7
.= (diff (f1,x0)) + (diff (f2,x0)) by A2, A6, A8, Def7 ;
:: thesis: verum