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, the carrier of S 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 A4: ( x1 in dom (p (#) f) & x2 in dom (p (#) f) ) ; :: thesis: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * |.(x1 - x2).|
A5: 0 <= |.(x1 - x2).| by COMPLEX1:46;
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| = ||.((p * (f /. x1)) - ((p (#) f) /. x2)).|| by A4, VFUNCT_1:def 4
.= ||.((p * (f /. x1)) - (p * (f /. x2))).|| by A4, VFUNCT_1:def 4
.= ||.((0. S) - (p * (f /. x2))).|| by A3, RLVECT_1:10
.= ||.((0. S) - (0. S)).|| by A3, RLVECT_1:10
.= 0 by NORMSP_1:6 ;
hence ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * |.(x1 - x2).| by A1, A5; :: thesis: verum

end;

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

end;

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

then 0 < |.p.| by COMPLEX1:47;
then A6: 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).| ) )
A7: 0 <= |.p.| by COMPLEX1:46;
thus 0 < g by A6; :: 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 A8: ( x1 in dom (p (#) f) & x2 in dom (p (#) f) ) ; :: thesis: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * |.(x1 - x2).|
then A9: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| = ||.((p * (f /. x1)) - ((p (#) f) /. x2)).|| by VFUNCT_1:def 4
.= ||.((p * (f /. x1)) - (p * (f /. x2))).|| by A8, VFUNCT_1:def 4
.= ||.(p * ((f /. x1) - (f /. x2))).|| by RLVECT_1:34
.= |.p.| * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def 1 ;
( x1 in dom f & x2 in dom f ) by A8, VFUNCT_1:def 4;
then |.p.| * ||.((f /. x1) - (f /. x2)).|| <= |.p.| * (s * |.(x1 - x2).|) by A2, A7, XREAL_1:64;
hence ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * |.(x1 - x2).| by A9; :: thesis: verum

end;

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

end;

end;