let x be set ; :: thesis: for h being PartFunc of REAL,REAL holds h | {x} is non-increasing

let h be PartFunc of REAL,REAL; :: thesis: h | {x} is non-increasing

let h be PartFunc of REAL,REAL; :: thesis: h | {x} is non-increasing

now :: thesis: for r1, r2 being Real st r1 in {x} /\ (dom h) & r2 in {x} /\ (dom h) & r1 < r2 holds

h . r2 <= h . r1

hence
h | {x} is non-increasing
by Th23; :: thesis: verumh . r2 <= h . r1

let r1, r2 be Real; :: thesis: ( r1 in {x} /\ (dom h) & r2 in {x} /\ (dom h) & r1 < r2 implies h . r2 <= h . r1 )

assume that

A1: r1 in {x} /\ (dom h) and

A2: r2 in {x} /\ (dom h) and

r1 < r2 ; :: thesis: h . r2 <= h . r1

r1 in {x} by A1, XBOOLE_0:def 4;

then A3: r1 = x by TARSKI:def 1;

r2 in {x} by A2, XBOOLE_0:def 4;

hence h . r2 <= h . r1 by A3, TARSKI:def 1; :: thesis: verum

end;assume that

A1: r1 in {x} /\ (dom h) and

A2: r2 in {x} /\ (dom h) and

r1 < r2 ; :: thesis: h . r2 <= h . r1

r1 in {x} by A1, XBOOLE_0:def 4;

then A3: r1 = x by TARSKI:def 1;

r2 in {x} by A2, XBOOLE_0:def 4;

hence h . r2 <= h . r1 by A3, TARSKI:def 1; :: thesis: verum