let X be set ; :: thesis: for p being Real
for f being PartFunc of REAL,REAL st X c= dom f & f | X is uniformly_continuous holds
(p (#) f) | X is uniformly_continuous
let p be Real; :: thesis: for f being PartFunc of REAL,REAL st X c= dom f & f | X is uniformly_continuous holds
(p (#) f) | X is uniformly_continuous
let f be PartFunc of REAL,REAL; :: thesis: ( X c= dom f & f | X is uniformly_continuous implies (p (#) f) | X is uniformly_continuous )
assume X c= dom f ; :: thesis: ( not f | X is uniformly_continuous or (p (#) f) | X is uniformly_continuous )
then A1: X c= dom (p (#) f) by VALUED_1:def 5;
assume A2: f | X is uniformly_continuous ; :: thesis: (p (#) f) | X is uniformly_continuous

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

suppose A3: p = 0 ; :: thesis: (p (#) f) | X is uniformly_continuous

now

let r be Real; :: thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s holds
abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r ) ) )
assume A4: 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s holds
abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r ) )
then consider s being Real such that
A5: 0 < s and
for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r by A2, Th1;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s holds
abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r ) )
thus 0 < s by A5; :: thesis: for x1, x2 being Real st x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s holds
abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r
let x1, x2 be Real; :: thesis: ( x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s implies abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r )
assume that
A6: x1 in dom ((p (#) f) | X) and
A7: x2 in dom ((p (#) f) | X) and
abs (x1 - x2) < s ; :: thesis: abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r
A8: x2 in X by A7, RELAT_1:57;
x1 in X by A6, RELAT_1:57;
then abs (((p (#) f) . x1) - ((p (#) f) . x2)) = abs ((p * (f . x1)) - ((p (#) f) . x2)) by A1, VALUED_1:def 5
.= abs (0 - (p * (f . x2))) by A1, A3, A8, VALUED_1:def 5
.= 0 by A3, ABSVALUE:2 ;
hence abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r by A4; :: thesis: verum

end;

hence (p (#) f) | X is uniformly_continuous by Th1; :: thesis: verum

end;

suppose A9: p <> 0 ; :: thesis: (p (#) f) | X is uniformly_continuous

then A10: 0 < abs p by COMPLEX1:47;
A11: 0 <> abs p by A9, COMPLEX1:47;

now

let r be Real; :: thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s holds
abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r ) ) )
assume 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s holds
abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r ) )
then 0 < r / (abs p) by A10, XREAL_1:139;
then consider s being Real such that
A12: 0 < s and
A13: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r / (abs p) by A2, Th1;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s holds
abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r ) )
thus 0 < s by A12; :: thesis: for x1, x2 being Real st x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s holds
abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r
let x1, x2 be Real; :: thesis: ( x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s implies abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r )
assume that
A14: x1 in dom ((p (#) f) | X) and
A15: x2 in dom ((p (#) f) | X) and
A16: abs (x1 - x2) < s ; :: thesis: abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r
A17: x2 in X by A15, RELAT_1:57;
A18: x1 in X by A14, RELAT_1:57;
then A19: abs (((p (#) f) . x1) - ((p (#) f) . x2)) = abs ((p * (f . x1)) - ((p (#) f) . x2)) by A1, VALUED_1:def 5
.= abs ((p * (f . x1)) - (p * (f . x2))) by A1, A17, VALUED_1:def 5
.= abs (p * ((f . x1) - (f . x2)))
.= (abs p) * (abs ((f . x1) - (f . x2))) by COMPLEX1:65 ;
x2 in dom (p (#) f) by A15, RELAT_1:57;
then x2 in dom f by VALUED_1:def 5;
then A20: x2 in dom (f | X) by A17, RELAT_1:57;
x1 in dom (p (#) f) by A14, RELAT_1:57;
then x1 in dom f by VALUED_1:def 5;
then x1 in dom (f | X) by A18, RELAT_1:57;
then (abs p) * (abs ((f . x1) - (f . x2))) < (r / (abs p)) * (abs p) by A10, A13, A16, A20, XREAL_1:68;
hence abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r by A11, A19, XCMPLX_1:87; :: thesis: verum

end;

hence (p (#) f) | X is uniformly_continuous by Th1; :: thesis: verum

end;

end;