let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL,REAL st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 . x0 holds
f2 * f1 is_continuous_in x0
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 . x0 implies f2 * f1 is_continuous_in x0 )
assume A1: x0 in dom (f2 * f1) ; :: thesis: ( not f1 is_continuous_in x0 or not f2 is_continuous_in f1 . x0 or f2 * f1 is_continuous_in x0 )
assume that
A2: f1 is_continuous_in x0 and
A3: f2 is_continuous_in f1 . x0 ; :: thesis: f2 * f1 is_continuous_in x0
let s1 be Real_Sequence; :: according to FCONT_1:def 1 :: thesis: ( rng s1 c= dom (f2 * f1) & s1 is convergent & lim s1 = x0 implies ( (f2 * f1) /* s1 is convergent & (f2 * f1) . x0 = lim ((f2 * f1) /* s1) ) )
assume that
A4: rng s1 c= dom (f2 * f1) and
A5: ( s1 is convergent & lim s1 = x0 ) ; :: thesis: ( (f2 * f1) /* s1 is convergent & (f2 * f1) . x0 = lim ((f2 * f1) /* s1) )
A6: dom (f2 * f1) c= dom f1 by RELAT_1:25;

now :: thesis: for x being object st x in rng (f1 /* s1) holds
x in dom f2

let x be object ; :: thesis: ( x in rng (f1 /* s1) implies x in dom f2 )
assume x in rng (f1 /* s1) ; :: thesis: x in dom f2
then consider n being Element of NAT such that
A7: x = (f1 /* s1) . n by FUNCT_2:113;
s1 . n in rng s1 by VALUED_0:28;
then f1 . (s1 . n) in dom f2 by A4, FUNCT_1:11;
hence x in dom f2 by A4, A6, A7, FUNCT_2:108, XBOOLE_1:1; :: thesis: verum

end;

then A8: rng (f1 /* s1) c= dom f2 ;

now :: thesis: for n being Element of NAT holds ((f2 * f1) /* s1) . n = (f2 /* (f1 /* s1)) . n

let n be Element of NAT ; :: thesis: ((f2 * f1) /* s1) . n = (f2 /* (f1 /* s1)) . n
s1 . n in rng s1 by VALUED_0:28;
then A9: s1 . n in dom f1 by A4, FUNCT_1:11;
thus ((f2 * f1) /* s1) . n = (f2 * f1) . (s1 . n) by A4, FUNCT_2:108
.= f2 . (f1 . (s1 . n)) by A9, FUNCT_1:13
.= f2 . ((f1 /* s1) . n) by A4, A6, FUNCT_2:108, XBOOLE_1:1
.= (f2 /* (f1 /* s1)) . n by A8, FUNCT_2:108 ; :: thesis: verum

end;

then A10: f2 /* (f1 /* s1) = (f2 * f1) /* s1 by FUNCT_2:63;
rng s1 c= dom f1 by A4, A6;
then A11: ( f1 /* s1 is convergent & f1 . x0 = lim (f1 /* s1) ) by A2, A5;
then f2 . (f1 . x0) = lim (f2 /* (f1 /* s1)) by A3, A8;
hence ( (f2 * f1) /* s1 is convergent & (f2 * f1) . x0 = lim ((f2 * f1) /* s1) ) by A1, A3, A11, A8, A10, FUNCT_1:12; :: thesis: verum