set X = dom f1;
set X1 = dom f2;
consider s being real number such that
A9: 0 < s and
A10: for x1, x2 being real number st x1 in dom (f1 | ((dom f1) /\ (dom f2))) & x2 in dom (f1 | ((dom f1) /\ (dom f2))) holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * (abs (x1 - x2)) by Th33;
consider g being real number such that
A11: 0 < g and
A12: for x1, x2 being real number st x1 in dom (f2 | ((dom f1) /\ (dom f2))) & x2 in dom (f2 | ((dom f1) /\ (dom f2))) holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * (abs (x1 - x2)) by Th33;

now

take p = s + g; :: thesis: ( 0 < p & ( for x1, x2 being real number st x1 in dom (f1 - f2) & x2 in dom (f1 - f2) holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2)) ) )
thus 0 < p by A9, A11; :: thesis: for x1, x2 being real number st x1 in dom (f1 - f2) & x2 in dom (f1 - f2) holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2))
let x1, x2 be real number ; :: thesis: ( x1 in dom (f1 - f2) & x2 in dom (f1 - f2) implies ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2)) )
assume A13: ( x1 in dom (f1 - f2) & x2 in dom (f1 - f2) ) ; :: thesis: ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2))
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| = ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 - f2) /. x2)).|| by A13, VFUNCT_1:def 2
.= ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 /. x2) - (f2 /. x2))).|| by A13, VFUNCT_1:def 2
.= ||.((f1 /. x1) - ((f2 /. x1) + ((f1 /. x2) - (f2 /. x2)))).|| by RLVECT_1:41
.= ||.((f1 /. x1) - (((f1 /. x2) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:42
.= ||.((f1 /. x1) - ((f1 /. x2) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:42
.= ||.(((f1 /. x1) - (f1 /. x2)) - ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:41 ;
then A15: ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by NORMSP_1:7;
dom (f2 | ((dom f1) /\ (dom f2))) = (dom f2) /\ ((dom f1) /\ (dom f2)) by RELAT_1:90
.= ((dom f2) /\ (dom f2)) /\ (dom f1) by XBOOLE_1:16
.= dom (f1 - f2) by VFUNCT_1:def 2 ;
then A16: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * (abs (x1 - x2)) by A12, A13;
dom (f1 | ((dom f1) /\ (dom f2))) = (dom f1) /\ ((dom f1) /\ (dom f2)) by RELAT_1:90
.= ((dom f1) /\ (dom f1)) /\ (dom f2) by XBOOLE_1:16
.= dom (f1 - f2) by VFUNCT_1:def 2 ;
then ||.((f1 /. x1) - (f1 /. x2)).|| <= s * (abs (x1 - x2)) by A10, A13;
then ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * (abs (x1 - x2))) + (g * (abs (x1 - x2))) by A16, XREAL_1:9;
hence ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2)) by A15, XXREAL_0:2; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL, the carrier of S st b₁ = f1 - f2 holds
b₁ is Lipschitzian by Def3; :: thesis: verum