let x0 be Real; :: thesis:
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis:
assume that
A1: f1 is_differentiable_in x0 and
A2: f2 is_differentiable_in x0 ; :: thesis:
consider N1 being Neighbourhood of x0 such that
A3: ( N1 c= dom f1 & ex L being LINEAR ex R being REST st
for x being Real st x in N1 holds
(f1 . x) - (f1 . x0) = (L . (x - x0)) + (R . (x - x0)) ) by A1, Def5;
consider L1 being LINEAR, R1 being REST such that
A4: for x being Real st x in N1 holds
(f1 . x) - (f1 . x0) = (L1 . (x - x0)) + (R1 . (x - x0)) by A3;
consider N2 being Neighbourhood of x0 such that
A5: ( N2 c= dom f2 & ex L being LINEAR ex R being REST st
for x being Real st x in N2 holds
(f2 . x) - (f2 . x0) = (L . (x - x0)) + (R . (x - x0)) ) by A2, Def5;
consider L2 being LINEAR, R2 being REST such that
A6: for x being Real st x in N2 holds
(f2 . x) - (f2 . x0) = (L2 . (x - x0)) + (R2 . (x - x0)) by A5;
consider N being Neighbourhood of x0 such that
A7: ( N c= N1 & N c= N2 ) by RCOMP_1:38;
reconsider L11 = (f2 . x0) (#) L1 as LINEAR by Th7;
reconsider L12 = (f1 . x0) (#) L2 as LINEAR by Th7;
A8: ( L11 is total & L12 is total & L1 is total & L2 is total ) by Def4;
reconsider L = L11 + L12 as LINEAR by Th6;
reconsider R11 = (f2 . x0) (#) R1 as REST by Th9;
reconsider R12 = (f1 . x0) (#) R2 as REST by Th9;
reconsider R13 = R11 + R12 as REST by Th8;
reconsider R14 = L1 (#) L2 as REST by Th10;
reconsider R15 = R13 + R14 as REST by Th8;
reconsider R16 = R1 (#) L2 as REST by Th11;
reconsider R17 = R1 (#) R2 as REST by Th8;
reconsider R18 = R2 (#) L1 as REST by Th11;
reconsider R19 = R16 + R17 as REST by Th8;
reconsider R20 = R19 + R18 as REST by Th8;
reconsider R = R15 + R20 as REST by Th8;
A9: ( R1 is total & R2 is total & R11 is total & R12 is total & R13 is total & R14 is total & R15 is total & R16 is total & R17 is total & R18 is total & R19 is total & R20 is total ) by Def3;
A10: N c= dom f1 by A3, A7, XBOOLE_1:1;
N c= dom f2 by A5, A7, XBOOLE_1:1;
then N /\ N c= (dom f1) /\ (dom f2) by A10, XBOOLE_1:27;
then A11: N c= dom (f1 (#) f2) by VALUED_1:def 4;

A12: now

let x be Real; :: thesis:
assume A13: x in N ; :: thesis:
then A14: ((f1 . x) - (f1 . x0)) + (f1 . x0) = ((L1 . (x - x0)) + (R1 . (x - x0))) + (f1 . x0) by A4, A7;
thus ((f1 (#) f2) . x) - ((f1 (#) f2) . x0) = ((f1 . x) * (f2 . x)) - ((f1 (#) f2) . x0) by VALUED_1:5
.= ((((f1 . x) * (f2 . x)) + (- ((f1 . x) * (f2 . x0)))) + ((f1 . x) * (f2 . x0))) - ((f1 . x0) * (f2 . x0)) by VALUED_1:5
.= ((f1 . x) * ((f2 . x) - (f2 . x0))) + (((f1 . x) - (f1 . x0)) * (f2 . x0))
.= ((f1 . x) * ((f2 . x) - (f2 . x0))) + (((L1 . (x - x0)) + (R1 . (x - x0))) * (f2 . x0)) by A4, A7, A13
.= ((f1 . x) * ((f2 . x) - (f2 . x0))) + (((f2 . x0) * (L1 . (x - x0))) + ((R1 . (x - x0)) * (f2 . x0)))
.= ((f1 . x) * ((f2 . x) - (f2 . x0))) + ((L11 . (x - x0)) + ((f2 . x0) * (R1 . (x - x0)))) by A8, RFUNCT_1:73
.= ((((L1 . (x - x0)) + (R1 . (x - x0))) + (f1 . x0)) * ((f2 . x) - (f2 . x0))) + ((L11 . (x - x0)) + (R11 . (x - x0))) by A9, A14, RFUNCT_1:73
.= ((((L1 . (x - x0)) + (R1 . (x - x0))) + (f1 . x0)) * ((L2 . (x - x0)) + (R2 . (x - x0)))) + ((L11 . (x - x0)) + (R11 . (x - x0))) by A6, A7, A13
.= ((((L1 . (x - x0)) + (R1 . (x - x0))) * ((L2 . (x - x0)) + (R2 . (x - x0)))) + (((f1 . x0) * (L2 . (x - x0))) + ((f1 . x0) * (R2 . (x - x0))))) + ((L11 . (x - x0)) + (R11 . (x - x0)))
.= ((((L1 . (x - x0)) + (R1 . (x - x0))) * ((L2 . (x - x0)) + (R2 . (x - x0)))) + ((L12 . (x - x0)) + ((f1 . x0) * (R2 . (x - x0))))) + ((L11 . (x - x0)) + (R11 . (x - x0))) by A8, RFUNCT_1:73
.= ((((L1 . (x - x0)) + (R1 . (x - x0))) * ((L2 . (x - x0)) + (R2 . (x - x0)))) + ((L12 . (x - x0)) + (R12 . (x - x0)))) + ((L11 . (x - x0)) + (R11 . (x - x0))) by A9, RFUNCT_1:73
.= (((L1 . (x - x0)) + (R1 . (x - x0))) * ((L2 . (x - x0)) + (R2 . (x - x0)))) + ((L12 . (x - x0)) + ((L11 . (x - x0)) + ((R11 . (x - x0)) + (R12 . (x - x0)))))
.= (((L1 . (x - x0)) + (R1 . (x - x0))) * ((L2 . (x - x0)) + (R2 . (x - x0)))) + ((L12 . (x - x0)) + ((L11 . (x - x0)) + (R13 . (x - x0)))) by A9, RFUNCT_1:72
.= (((L1 . (x - x0)) + (R1 . (x - x0))) * ((L2 . (x - x0)) + (R2 . (x - x0)))) + (((L11 . (x - x0)) + (L12 . (x - x0))) + (R13 . (x - x0)))
.= ((((L1 . (x - x0)) * (L2 . (x - x0))) + ((L1 . (x - x0)) * (R2 . (x - x0)))) + ((R1 . (x - x0)) * ((L2 . (x - x0)) + (R2 . (x - x0))))) + ((L . (x - x0)) + (R13 . (x - x0))) by A8, RFUNCT_1:72
.= (((R14 . (x - x0)) + ((R2 . (x - x0)) * (L1 . (x - x0)))) + ((R1 . (x - x0)) * ((L2 . (x - x0)) + (R2 . (x - x0))))) + ((L . (x - x0)) + (R13 . (x - x0))) by A8, RFUNCT_1:72
.= (((R14 . (x - x0)) + (R18 . (x - x0))) + (((R1 . (x - x0)) * (L2 . (x - x0))) + ((R1 . (x - x0)) * (R2 . (x - x0))))) + ((L . (x - x0)) + (R13 . (x - x0))) by A8, A9, RFUNCT_1:72
.= (((R14 . (x - x0)) + (R18 . (x - x0))) + ((R16 . (x - x0)) + ((R1 . (x - x0)) * (R2 . (x - x0))))) + ((L . (x - x0)) + (R13 . (x - x0))) by A8, A9, RFUNCT_1:72
.= (((R14 . (x - x0)) + (R18 . (x - x0))) + ((R16 . (x - x0)) + (R17 . (x - x0)))) + ((L . (x - x0)) + (R13 . (x - x0))) by A9, RFUNCT_1:72
.= (((R14 . (x - x0)) + (R18 . (x - x0))) + (R19 . (x - x0))) + ((L . (x - x0)) + (R13 . (x - x0))) by A9, RFUNCT_1:72
.= ((R14 . (x - x0)) + ((R19 . (x - x0)) + (R18 . (x - x0)))) + ((L . (x - x0)) + (R13 . (x - x0)))
.= ((L . (x - x0)) + (R13 . (x - x0))) + ((R14 . (x - x0)) + (R20 . (x - x0))) by A9, RFUNCT_1:72
.= (L . (x - x0)) + (((R13 . (x - x0)) + (R14 . (x - x0))) + (R20 . (x - x0)))
.= (L . (x - x0)) + ((R15 . (x - x0)) + (R20 . (x - x0))) by A9, RFUNCT_1:72
.= (L . (x - x0)) + (R . (x - x0)) by A9, RFUNCT_1:72 ; :: thesis:

end;

hence f1 (#) f2 is_differentiable_in x0 by A11, Def5; :: thesis:
hence diff (f1 (#) f2),x0 = L . 1 by A11, A12, Def6
.= (L11 . 1) + (L12 . 1) by A8, RFUNCT_1:72
.= ((f2 . x0) * (L1 . 1)) + (L12 . 1) by A8, RFUNCT_1:73
.= ((f2 . x0) * (L1 . 1)) + ((f1 . x0) * (L2 . 1)) by A8, RFUNCT_1:73
.= ((f2 . x0) * (diff f1,x0)) + ((f1 . x0) * (L2 . 1)) by A1, A3, A4, Def6
.= ((f2 . x0) * (diff f1,x0)) + ((f1 . x0) * (diff f2,x0)) by A2, A5, A6, Def6 ;
:: thesis: