let r be Real; :: thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
let CNS be ComplexNormSpace; :: thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
let RNS be RealNormSpace; :: thesis: for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
let X be set ; :: thesis: for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
let f be PartFunc of CNS,RNS; :: thesis: ( f is_Lipschitzian_on X implies r (#) f is_Lipschitzian_on X )
assume A1: f is_Lipschitzian_on X ; :: thesis: r (#) f is_Lipschitzian_on X
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ;
X c= dom f by A1;
hence A4: X c= dom (r (#) f) by VFUNCT_1:def 4; :: according to NCFCONT1:def 18 :: thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) )

now :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )

per cases ( r = 0 or r <> 0 ) ;

suppose A5: r = 0 ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )

take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A2; :: thesis: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of CNS; :: thesis: ( x1 in X & x2 in X implies ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| )
assume that
A6: x1 in X and
A7: x2 in X ; :: thesis: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
0 <= ||.(x1 - x2).|| by CLVECT_1:105;
then A8: s * 0 <= s * ||.(x1 - x2).|| by A2;
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| = ||.((r * (f /. x1)) - ((r (#) f) /. x2)).|| by A4, A6, VFUNCT_1:def 4
.= ||.((0. RNS) - ((r (#) f) /. x2)).|| by A5, RLVECT_1:10
.= ||.((0. RNS) - (r * (f /. x2))).|| by A4, A7, VFUNCT_1:def 4
.= ||.((0. RNS) - (0. RNS)).|| by A5, RLVECT_1:10
.= ||.(0. RNS).|| by RLVECT_1:13
.= 0 by NORMSP_1:1 ;
hence ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| by A8; :: thesis: verum

end;

suppose A9: r <> 0 ; :: thesis: ex g being Real st
( 0 < g & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )

reconsider g = |.r.| * s as Real ;
take g = g; :: thesis: ( 0 < g & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
0 < |.r.| by A9, COMPLEX1:47;
then 0 * s < |.r.| * s by A2, XREAL_1:68;
hence 0 < g ; :: thesis: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
let x1, x2 be Point of CNS; :: thesis: ( x1 in X & x2 in X implies ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| )
assume that
A10: x1 in X and
A11: x2 in X ; :: thesis: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
0 <= |.r.| by COMPLEX1:46;
then A12: |.r.| * ||.((f /. x1) - (f /. x2)).|| <= |.r.| * (s * ||.(x1 - x2).||) by A3, A10, A11, XREAL_1:64;
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| = ||.((r * (f /. x1)) - ((r (#) f) /. x2)).|| by A4, A10, VFUNCT_1:def 4
.= ||.((r * (f /. x1)) - (r * (f /. x2))).|| by A4, A11, VFUNCT_1:def 4
.= ||.(r * ((f /. x1) - (f /. x2))).|| by RLVECT_1:34
.= |.r.| * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def 1 ;
hence ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| by A12; :: thesis: verum

end;

end;

end;

hence ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ; :: thesis: verum