let X be set ; :: thesis: for f1, f2 being PartFunc of REAL,REAL st X c= (dom f1) /\ (dom f2) & f1 | X is continuous & f2 | X is continuous holds
( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous & (f1 (#) f2) | X is continuous )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( X c= (dom f1) /\ (dom f2) & f1 | X is continuous & f2 | X is continuous implies ( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous & (f1 (#) f2) | X is continuous ) )
assume A1: X c= (dom f1) /\ (dom f2) ; :: thesis: ( not f1 | X is continuous or not f2 | X is continuous or ( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous & (f1 (#) f2) | X is continuous ) )
assume A2: ( f1 | X is continuous & f2 | X is continuous ) ; :: thesis: ( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous & (f1 (#) f2) | X is continuous )
A3: ( X c= dom f1 & X c= dom f2 ) by A1, XBOOLE_1:18;

A4: now :: thesis: for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds
( (f1 (#) f2) /* s1 is convergent & (f1 (#) f2) . (lim s1) = lim ((f1 (#) f2) /* s1) )

let s1 be Real_Sequence; :: thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 (#) f2) /* s1 is convergent & (f1 (#) f2) . (lim s1) = lim ((f1 (#) f2) /* s1) ) )
assume that
A5: rng s1 c= X and
A6: ( s1 is convergent & lim s1 in X ) ; :: thesis: ( (f1 (#) f2) /* s1 is convergent & (f1 (#) f2) . (lim s1) = lim ((f1 (#) f2) /* s1) )
A7: rng s1 c= (dom f1) /\ (dom f2) by A1, A5;
A8: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A3, A2, A5, A6, Th13;
then A9: (f1 /* s1) (#) (f2 /* s1) is convergent ;
( f1 . (lim s1) = lim (f1 /* s1) & f2 . (lim s1) = lim (f2 /* s1) ) by A3, A2, A5, A6, Th13;
then (f1 (#) f2) . (lim s1) = (lim (f1 /* s1)) * (lim (f2 /* s1)) by VALUED_1:5
.= lim ((f1 /* s1) (#) (f2 /* s1)) by A8, SEQ_2:15
.= lim ((f1 (#) f2) /* s1) by A7, RFUNCT_2:8 ;
hence ( (f1 (#) f2) /* s1 is convergent & (f1 (#) f2) . (lim s1) = lim ((f1 (#) f2) /* s1) ) by A7, A9, RFUNCT_2:8; :: thesis: verum

end;

A10: X c= dom (f1 + f2) by A1, VALUED_1:def 1;

now :: thesis: for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds
( (f1 + f2) /* s1 is convergent & (f1 + f2) . (lim s1) = lim ((f1 + f2) /* s1) )

let s1 be Real_Sequence; :: thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) . (lim s1) = lim ((f1 + f2) /* s1) ) )
assume that
A11: rng s1 c= X and
A12: s1 is convergent and
A13: lim s1 in X ; :: thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) . (lim s1) = lim ((f1 + f2) /* s1) )
A14: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A3, A2, A11, A12, A13, Th13;
then A15: (f1 /* s1) + (f2 /* s1) is convergent ;
A16: rng s1 c= (dom f1) /\ (dom f2) by A1, A11;
( f1 . (lim s1) = lim (f1 /* s1) & f2 . (lim s1) = lim (f2 /* s1) ) by A3, A2, A11, A12, A13, Th13;
then (f1 + f2) . (lim s1) = (lim (f1 /* s1)) + (lim (f2 /* s1)) by A10, A13, VALUED_1:def 1
.= lim ((f1 /* s1) + (f2 /* s1)) by A14, SEQ_2:6
.= lim ((f1 + f2) /* s1) by A16, RFUNCT_2:8 ;
hence ( (f1 + f2) /* s1 is convergent & (f1 + f2) . (lim s1) = lim ((f1 + f2) /* s1) ) by A16, A15, RFUNCT_2:8; :: thesis: verum

end;

hence (f1 + f2) | X is continuous by A10, Th13; :: thesis: ( (f1 - f2) | X is continuous & (f1 (#) f2) | X is continuous )
A17: X c= dom (f1 - f2) by A1, VALUED_1:12;

now :: thesis: for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds
( (f1 - f2) /* s1 is convergent & (f1 - f2) . (lim s1) = lim ((f1 - f2) /* s1) )

let s1 be Real_Sequence; :: thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) . (lim s1) = lim ((f1 - f2) /* s1) ) )
assume that
A18: rng s1 c= X and
A19: s1 is convergent and
A20: lim s1 in X ; :: thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) . (lim s1) = lim ((f1 - f2) /* s1) )
A21: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A3, A2, A18, A19, A20, Th13;
then A22: (f1 /* s1) - (f2 /* s1) is convergent ;
A23: rng s1 c= (dom f1) /\ (dom f2) by A1, A18;
( f1 . (lim s1) = lim (f1 /* s1) & f2 . (lim s1) = lim (f2 /* s1) ) by A3, A2, A18, A19, A20, Th13;
then (f1 - f2) . (lim s1) = (lim (f1 /* s1)) - (lim (f2 /* s1)) by A17, A20, VALUED_1:13
.= lim ((f1 /* s1) - (f2 /* s1)) by A21, SEQ_2:12
.= lim ((f1 - f2) /* s1) by A23, RFUNCT_2:8 ;
hence ( (f1 - f2) /* s1 is convergent & (f1 - f2) . (lim s1) = lim ((f1 - f2) /* s1) ) by A23, A22, RFUNCT_2:8; :: thesis: verum

end;

hence (f1 - f2) | X is continuous by A17, Th13; :: thesis: (f1 (#) f2) | X is continuous
X c= dom (f1 (#) f2) by A1, VALUED_1:def 4;
hence (f1 (#) f2) | X is continuous by A4, Th13; :: thesis: verum