set X = dom f1;
set X1 = dom f2;
consider s being Real such that
A1: 0 < s and
A2: for x1, x2 being Real st x1 in dom (f1 | ((dom f1) /\ (dom f2))) & x2 in dom (f1 | ((dom f1) /\ (dom f2))) holds
|.((f1 . x1) - (f1 . x2)).| <= s * |.(x1 - x2).| by Th32;
consider g being Real such that
A3: 0 < g and
A4: for x1, x2 being Real st x1 in dom (f2 | ((dom f1) /\ (dom f2))) & x2 in dom (f2 | ((dom f1) /\ (dom f2))) holds
|.((f2 . x1) - (f2 . x2)).| <= g * |.(x1 - x2).| by Th32;

now :: thesis: ex p being set st
( 0 < p & ( for x1, x2 being Real st x1 in dom (f1 + f2) & x2 in dom (f1 + f2) holds
|.(((f1 + f2) . x1) - ((f1 + f2) . x2)).| <= p * |.(x1 - x2).| ) )

take p = s + g; :: thesis: ( 0 < p & ( for x1, x2 being Real st x1 in dom (f1 + f2) & x2 in dom (f1 + f2) holds
|.(((f1 + f2) . x1) - ((f1 + f2) . x2)).| <= p * |.(x1 - x2).| ) )
thus 0 < p by A1, A3; :: thesis: for x1, x2 being Real st x1 in dom (f1 + f2) & x2 in dom (f1 + f2) holds
|.(((f1 + f2) . x1) - ((f1 + f2) . x2)).| <= p * |.(x1 - x2).|
let x1, x2 be Real; :: thesis: ( x1 in dom (f1 + f2) & x2 in dom (f1 + f2) implies |.(((f1 + f2) . x1) - ((f1 + f2) . x2)).| <= p * |.(x1 - x2).| )
assume that
A5: x1 in dom (f1 + f2) and
A6: x2 in dom (f1 + f2) ; :: thesis: |.(((f1 + f2) . x1) - ((f1 + f2) . x2)).| <= p * |.(x1 - x2).|
|.(((f1 + f2) . x1) - ((f1 + f2) . x2)).| = |.(((f1 . x1) + (f2 . x1)) - ((f1 + f2) . x2)).| by A5, VALUED_1:def 1
.= |.(((f1 . x1) + (f2 . x1)) - ((f1 . x2) + (f2 . x2))).| by A6, VALUED_1:def 1
.= |.(((f1 . x1) - (f1 . x2)) + ((f2 . x1) - (f2 . x2))).| ;
then A7: |.(((f1 + f2) . x1) - ((f1 + f2) . x2)).| <= |.((f1 . x1) - (f1 . x2)).| + |.((f2 . x1) - (f2 . x2)).| by COMPLEX1:56;
dom (f2 | ((dom f1) /\ (dom f2))) = (dom f2) /\ ((dom f1) /\ (dom f2)) by RELAT_1:61
.= ((dom f2) /\ (dom f2)) /\ (dom f1) by XBOOLE_1:16
.= dom (f1 + f2) by VALUED_1:def 1 ;
then A8: |.((f2 . x1) - (f2 . x2)).| <= g * |.(x1 - x2).| by A4, A5, A6;
dom (f1 | ((dom f1) /\ (dom f2))) = (dom f1) /\ ((dom f1) /\ (dom f2)) by RELAT_1:61
.= ((dom f1) /\ (dom f1)) /\ (dom f2) by XBOOLE_1:16
.= dom (f1 + f2) by VALUED_1:def 1 ;
then |.((f1 . x1) - (f1 . x2)).| <= s * |.(x1 - x2).| by A2, A5, A6;
then |.((f1 . x1) - (f1 . x2)).| + |.((f2 . x1) - (f2 . x2)).| <= (s * |.(x1 - x2).|) + (g * |.(x1 - x2).|) by A8, XREAL_1:7;
hence |.(((f1 + f2) . x1) - ((f1 + f2) . x2)).| <= p * |.(x1 - x2).| by A7, XXREAL_0:2; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL,REAL st b₁ = f1 + f2 holds
b₁ is Lipschitzian ; :: thesis: verum