consider s being Real such that
A1: 0 < s and
A2: for x1, x2 being Real st x1 in dom f & x2 in dom f holds
|.((f . x1) - (f . x2)).| <= s * |.(x1 - x2).| by Def3;

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

suppose A3: p = 0 ; :: thesis: for b₁ being PartFunc of REAL,REAL st b₁ = p (#) f holds
b₁ is Lipschitzian

now :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds
|.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= s * |.(x1 - x2).| ) )

take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds
|.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= s * |.(x1 - x2).| ) )
thus 0 < s by A1; :: thesis: for x1, x2 being Real st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds
|.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= s * |.(x1 - x2).|
let x1, x2 be Real; :: thesis: ( x1 in dom (p (#) f) & x2 in dom (p (#) f) implies |.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= s * |.(x1 - x2).| )
assume that
A4: x1 in dom (p (#) f) and
A5: x2 in dom (p (#) f) ; :: thesis: |.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= s * |.(x1 - x2).|
A6: 0 <= |.(x1 - x2).| by COMPLEX1:46;
|.(((p (#) f) . x1) - ((p (#) f) . x2)).| = |.((p * (f . x1)) - ((p (#) f) . x2)).| by A4, VALUED_1:def 5
.= |.(0 - (p * (f . x2))).| by A3, A5, VALUED_1:def 5
.= 0 by A3, ABSVALUE:2 ;
hence |.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= s * |.(x1 - x2).| by A1, A6; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL,REAL st b₁ = p (#) f holds
b₁ is Lipschitzian ; :: thesis: verum

end;

suppose p <> 0 ; :: thesis: for b₁ being PartFunc of REAL,REAL st b₁ = p (#) f holds
b₁ is Lipschitzian

then 0 < |.p.| by COMPLEX1:47;
then A7: 0 * s < |.p.| * s by A1, XREAL_1:68;

now :: thesis: ex g being set st
( 0 < g & ( for x1, x2 being Real st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds
|.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= g * |.(x1 - x2).| ) )

take g = |.p.| * s; :: thesis: ( 0 < g & ( for x1, x2 being Real st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds
|.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= g * |.(x1 - x2).| ) )
A8: 0 <= |.p.| by COMPLEX1:46;
thus 0 < g by A7; :: thesis: for x1, x2 being Real st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds
|.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= g * |.(x1 - x2).|
let x1, x2 be Real; :: thesis: ( x1 in dom (p (#) f) & x2 in dom (p (#) f) implies |.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= g * |.(x1 - x2).| )
assume that
A9: x1 in dom (p (#) f) and
A10: x2 in dom (p (#) f) ; :: thesis: |.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= g * |.(x1 - x2).|
A11: |.(((p (#) f) . x1) - ((p (#) f) . x2)).| = |.((p * (f . x1)) - ((p (#) f) . x2)).| by A9, VALUED_1:def 5
.= |.((p * (f . x1)) - (p * (f . x2))).| by A10, VALUED_1:def 5
.= |.(p * ((f . x1) - (f . x2))).|
.= |.p.| * |.((f . x1) - (f . x2)).| by COMPLEX1:65 ;
( x1 in dom f & x2 in dom f ) by A9, A10, VALUED_1:def 5;
then |.p.| * |.((f . x1) - (f . x2)).| <= |.p.| * (s * |.(x1 - x2).|) by A2, A8, XREAL_1:64;
hence |.(((p (#) f) . x1) - ((p (#) f) . x2)).| <= g * |.(x1 - x2).| by A11; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL,REAL st b₁ = p (#) f holds
b₁ is Lipschitzian ; :: thesis: verum

end;

end;