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 X c= dom f by Def28;
hence A2: X c= dom (r (#) f) by VFUNCT_1:def 4; :: according to NCFCONT1:def 28 :: 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).|| ) )
consider s being Real such that
A3: ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ) ) by A1, Def28;

now

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

suppose A4: 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 A3; :: 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 A5: ( x1 in X & x2 in X ) ; :: thesis: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
then A6: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| = ||.((r * (f /. x1)) - ((r (#) f) /. x2)).|| by A2, VFUNCT_1:def 4
.= ||.((0. RNS) - ((r (#) f) /. x2)).|| by A4, RLVECT_1:23
.= ||.((0. RNS) - (r * (f /. x2))).|| by A2, A5, VFUNCT_1:def 4
.= ||.((0. RNS) - (0. RNS)).|| by A4, RLVECT_1:23
.= ||.(0. RNS).|| by RLVECT_1:26
.= 0 by NORMSP_1:5 ;
0 <= ||.(x1 - x2).|| by CLVECT_1:106;
then s * 0 <= s * ||.(x1 - x2).|| by A3, XREAL_1:66;
hence ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| by A6; :: thesis: verum

end;

suppose r <> 0 ; :: thesis: ex g being Element of 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).|| ) )

then 0 < abs r by COMPLEX1:133;
then A7: 0 * s < (abs r) * s by A3, XREAL_1:70;
take g = (abs r) * s; :: 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).|| ) )
thus 0 < g by A7; :: 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 A8: ( x1 in X & x2 in X ) ; :: thesis: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
then A9: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| = ||.((r * (f /. x1)) - ((r (#) f) /. x2)).|| by A2, VFUNCT_1:def 4
.= ||.((r * (f /. x1)) - (r * (f /. x2))).|| by A2, A8, VFUNCT_1:def 4
.= ||.(r * ((f /. x1) - (f /. x2))).|| by RLVECT_1:48
.= (abs r) * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def 2 ;
0 <= abs r by COMPLEX1:132;
then (abs r) * ||.((f /. x1) - (f /. x2)).|| <= (abs r) * (s * ||.(x1 - x2).||) by A3, A8, XREAL_1:66;
hence ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| by A9; :: 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