let x be set ; :: thesis: for h being PartFunc of REAL,REAL holds h | {x} is decreasing
let h be PartFunc of REAL,REAL; :: thesis: h | {x} is 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
let r1, r2 be Real; :: thesis: ( r1 in {x} /\ (dom h) & r2 in {x} /\ (dom h) & r1 < r2 implies h . r1 > h . r2 )
assume that
A1: r1 in {x} /\ (dom h) and
A2: r2 in {x} /\ (dom h) and
A3: r1 < r2 ; :: thesis: h . r1 > h . r2
r1 in {x} by A1, XBOOLE_0:def 4;
then A4: r1 = x by TARSKI:def 1;
r2 in {x} by A2, XBOOLE_0:def 4;
hence h . r1 > h . r2 by A3, A4, TARSKI:def 1; :: thesis: verum
end;
hence h | {x} is decreasing by Th21; :: thesis: verum