let CNS be ComplexNormSpace; :: thesis: for RNS being RealNormSpace
for X being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let RNS be RealNormSpace; :: thesis: for X being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let X be set ; :: thesis: for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let f1, f2 be PartFunc of CNS,RNS; :: thesis: ( f1 is_continuous_on X & f2 is_continuous_on X implies ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X ) )
assume A1: ( f1 is_continuous_on X & f2 is_continuous_on X ) ; :: thesis: ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
then ( X c= dom f1 & X c= dom f2 ) ;
then A2: X c= (dom f1) /\ (dom f2) by XBOOLE_1:19;
then A3: X c= dom (f1 + f2) by VFUNCT_1:def 1;

now :: thesis: for s1 being sequence of CNS 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 sequence of CNS; :: 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
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; :: thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) )
A7: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A4, A5, A6, Th42;
then A8: (f1 /* s1) + (f2 /* s1) is convergent by NORMSP_1:19;
A9: rng s1 c= (dom f1) /\ (dom f2) by A2, A4;
( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A4, A5, A6, Th42;
then (f1 + f2) /. (lim s1) = (lim (f1 /* s1)) + (lim (f2 /* s1)) by A3, A6, VFUNCT_1:def 1
.= lim ((f1 /* s1) + (f2 /* s1)) by A7, NORMSP_1:25
.= lim ((f1 + f2) /* s1) by A9, Th24 ;
hence ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) by A9, A8, Th24; :: thesis: verum

end;

hence f1 + f2 is_continuous_on X by A3, Th42; :: thesis: f1 - f2 is_continuous_on X
A10: X c= dom (f1 - f2) by A2, VFUNCT_1:def 2;

now :: thesis: for s1 being sequence of CNS 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 sequence of CNS; :: 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 A1, A11, A12, A13, Th42;
then A15: (f1 /* s1) - (f2 /* s1) is convergent by NORMSP_1:20;
A16: rng s1 c= (dom f1) /\ (dom f2) by A2, A11;
( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A11, A12, A13, Th42;
then (f1 - f2) /. (lim s1) = (lim (f1 /* s1)) - (lim (f2 /* s1)) by A10, A13, VFUNCT_1:def 2
.= lim ((f1 /* s1) - (f2 /* s1)) by A14, NORMSP_1:26
.= lim ((f1 - f2) /* s1) by A16, Th24 ;
hence ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) by A16, A15, Th24; :: thesis: verum

end;

hence f1 - f2 is_continuous_on X by A10, Th42; :: thesis: verum