reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  f1|X is Lipschitzian & f2|X1 is Lipschitzian & f1|Z is bounded & f2|Z1
  is bounded implies (f1(#)f2)|(X /\ Z /\ X1 /\ Z1) is Lipschitzian
proof
A1: f1|(X /\ Z /\ X1 /\ Z1) = f1|(X1 /\ Z1 /\ (X /\ Z)) by XBOOLE_1:16
    .= f1|(X1 /\ Z1 /\ X /\ Z) by XBOOLE_1:16
    .= f1|Z|(X1 /\ Z1 /\ X) by RELAT_1:71;
A2: f1|(X /\ Z /\ X1 /\ Z1) = f1|(X1 /\ Z1 /\ (X /\ Z)) by XBOOLE_1:16
    .= f1|(X1 /\ Z1 /\ Z /\ X) by XBOOLE_1:16
    .= f1|X|(X1 /\ Z1 /\ Z) by RELAT_1:71;
A3: f2|(X /\ Z /\ X1 /\ Z1) = f2|(X /\ Z /\ Z1 /\ X1) by XBOOLE_1:16
    .= f2|X1|(Z /\ X /\ Z1) by RELAT_1:71;
A4: (f1(#)f2)|(X /\ Z /\ X1 /\ Z1) = f1|(X /\ Z /\ X1 /\ Z1)(#)f2|(X /\ Z /\
  X1 /\ Z1) & f2|(X /\ Z /\ X1 /\ Z1) = f2|Z1|(X /\ Z /\ X1) by RELAT_1:71
,RFUNCT_1:45;
  assume f1|X is Lipschitzian & f2|X1 is Lipschitzian & f1|Z is bounded & f2|
  Z1 is bounded;
  hence thesis by A1,A2,A4,A3;
end;
