let X, Y be set ; :: thesis: for h being PartFunc of REAL,REAL st X c= Y & h | Y is non-decreasing holds
h | X is non-decreasing
let h be PartFunc of REAL,REAL; :: thesis: ( X c= Y & h | Y is non-decreasing implies h | X is non-decreasing )
assume that
A1: X c= Y and
A2: h | Y is non-decreasing ; :: thesis: h | X is non-decreasing
now :: thesis: for r1, r2 being Real st r1 in X /\ (dom h) & r2 in X /\ (dom h) & r1 < r2 holds
h . r1 <= h . r2
A3: X /\ (dom h) c= Y /\ (dom h) by A1, XBOOLE_1:26;
let r1, r2 be Real; :: thesis: ( r1 in X /\ (dom h) & r2 in X /\ (dom h) & r1 < r2 implies h . r1 <= h . r2 )
assume ( r1 in X /\ (dom h) & r2 in X /\ (dom h) & r1 < r2 ) ; :: thesis: h . r1 <= h . r2
hence h . r1 <= h . r2 by A2, A3, Th22; :: thesis: verum
end;
hence h | X is non-decreasing by Th22; :: thesis: verum