let x0, r be Real; :: thesis: for f being PartFunc of REAL,REAL st f is_differentiable_in x0 holds
( r (#) f is_differentiable_in x0 & diff ((r (#) f),x0) = r * (diff (f,x0)) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_differentiable_in x0 implies ( r (#) f is_differentiable_in x0 & diff ((r (#) f),x0) = r * (diff (f,x0)) ) )
reconsider j = 1 as Element of REAL by XREAL_0:def 1;
assume A1: f is_differentiable_in x0 ; :: thesis: ( r (#) f is_differentiable_in x0 & diff ((r (#) f),x0) = r * (diff (f,x0)) )
then consider N1 being Neighbourhood of x0 such that
A2: N1 c= dom f and
A3: ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N1 holds
(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) ;
consider L1 being LinearFunc, R1 being RestFunc such that
A4: for x being Real st x in N1 holds
(f . x) - (f . x0) = (L1 . (x - x0)) + (R1 . (x - x0)) by A3;
reconsider R = r (#) R1 as RestFunc by Th5;
reconsider L = r (#) L1 as LinearFunc by Th3;
A5: L1 is total by Def3;
A6: N1 c= dom (r (#) f) by A2, VALUED_1:def 5;
A7: R1 is total by Def2;

A8: now :: thesis: for x being Real st x in N1 holds
((r (#) f) . x) - ((r (#) f) . x0) = (L . (x - x0)) + (R . (x - x0))

let x be Real; :: thesis: ( x in N1 implies ((r (#) f) . x) - ((r (#) f) . x0) = (L . (x - x0)) + (R . (x - x0)) )
A9: x0 in N1 by RCOMP_1:16;
reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;
assume A10: x in N1 ; :: thesis: ((r (#) f) . x) - ((r (#) f) . x0) = (L . (x - x0)) + (R . (x - x0))
hence ((r (#) f) . x) - ((r (#) f) . x0) = (r * (f . x)) - ((r (#) f) . x0) by A6, VALUED_1:def 5
.= (r * (f . x)) - (r * (f . x0)) by A6, A9, VALUED_1:def 5
.= r * ((f . x) - (f . x0))
.= r * ((L1 . (x - x0)) + (R1 . (x - x0))) by A4, A10
.= (r * (L1 . (x - x0))) + (r * (R1 . (x - x0)))
.= (L . (xx - xx0)) + (r * (R1 . (xx - xx0))) by A5, RFUNCT_1:57
.= (L . (x - x0)) + (R . (x - x0)) by A7, RFUNCT_1:57 ;
:: thesis: verum

end;

hence r (#) f is_differentiable_in x0 by A6; :: thesis: diff ((r (#) f),x0) = r * (diff (f,x0))
hence diff ((r (#) f),x0) = L . 1 by A6, A8, Def5
.= r * (L1 . j) by A5, RFUNCT_1:57
.= r * (diff (f,x0)) by A1, A2, A4, Def5 ;
:: thesis: verum