let X, X1 be set ; :: thesis: for S, T being RealNormSpace
for f1, f2 being PartFunc of S,T st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let S, T be RealNormSpace; :: thesis: for f1, f2 being PartFunc of S,T st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of S,T; :: thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 + f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; :: thesis: f1 + f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th38, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| ;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th38, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 ;
X /\ X1 c= dom f1 by A3;
then X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 + f2) by VFUNCT_1:def 1; :: according to NFCONT_1:def 9 :: thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6;
take p = s + g; :: thesis: ( 0 < p & ( for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
0 + 0 < s + g by A4, A9;
hence 0 < p ; :: thesis: for x1, x2 being Point of S st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of S; :: thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; :: thesis: ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| = ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 + f2) /. x2)).|| by A8, A11, VFUNCT_1:def 1
.= ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 /. x2) + (f2 /. x2))).|| by A8, A12, VFUNCT_1:def 1
.= ||.((f1 /. x1) + ((f2 /. x1) - ((f1 /. x2) + (f2 /. x2)))).|| by RLVECT_1:def 3
.= ||.((f1 /. x1) + (((f2 /. x1) - (f1 /. x2)) - (f2 /. x2))).|| by RLVECT_1:27
.= ||.((f1 /. x1) + (((- (f1 /. x2)) + (f2 /. x1)) - (f2 /. x2))).||
.= ||.((f1 /. x1) + ((- (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:def 3
.= ||.(((f1 /. x1) - (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def 3 ;
then ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by NORMSP_1:def 1;
hence ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; :: thesis: verum