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 A2: X c= dom (p (#) f) by VALUED_1:def 5;
assume A1: 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 & ( 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 A1, Def1;
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 ( x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s ) ; :: thesis: abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r
then B6: ( x1 in X & x2 in X ) by RELAT_1:86;
then abs (((p (#) f) . x1) - ((p (#) f) . x2)) = abs ((p * (f . x1)) - ((p (#) f) . x2)) by A2, VALUED_1:def 5
.= abs (0 - (p * (f . x2))) by A2, A3, B6, VALUED_1:def 5
.= 0 by A3, ABSVALUE:7 ;
hence abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r by A4; :: thesis: verum

end;

hence (p (#) f) | X is uniformly_continuous by Def1; :: thesis: verum

end;

suppose A7: p <> 0 ; :: thesis: (p (#) f) | X is uniformly_continuous

then A8: 0 < abs p by COMPLEX1:133;
A9: 0 <> abs p by A7, COMPLEX1:133;

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 A8, XREAL_1:141;
then consider s being Real such that
A10: ( 0 < s & ( 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 A1, Def1;
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 A10; :: 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 A11: ( x1 in dom ((p (#) f) | X) & x2 in dom ((p (#) f) | X) & abs (x1 - x2) < s ) ; :: thesis: abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r
then B6: ( x1 in X & x2 in X ) by RELAT_1:86;
( x1 in dom (p (#) f) & x2 in dom (p (#) f) ) by A11, RELAT_1:86;
then ( x1 in dom f & x2 in dom f ) by VALUED_1:def 5;
then C6: ( x1 in dom (f | X) & x2 in dom (f | X) ) by B6, RELAT_1:86;
A12: abs (((p (#) f) . x1) - ((p (#) f) . x2)) = abs ((p * (f . x1)) - ((p (#) f) . x2)) by A2, B6, VALUED_1:def 5
.= abs ((p * (f . x1)) - (p * (f . x2))) by A2, B6, VALUED_1:def 5
.= abs (p * ((f . x1) - (f . x2)))
.= (abs p) * (abs ((f . x1) - (f . x2))) by COMPLEX1:151 ;
(abs p) * (abs ((f . x1) - (f . x2))) < (r / (abs p)) * (abs p) by A8, A10, A11, C6, XREAL_1:70;
hence abs (((p (#) f) . x1) - ((p (#) f) . x2)) < r by A9, A12, XCMPLX_1:88; :: thesis: verum

end;

hence (p (#) f) | X is uniformly_continuous by Def1; :: thesis: verum

end;

end;