thus ( h is non-increasing implies for r1, r2 being Real st r1 in dom h & r2 in dom h & r1 < r2 holds
h . r1 >= h . r2 ) ; :: thesis: ( ( for r1, r2 being Real st r1 in dom h & r2 in dom h & r1 < r2 holds
h . r2 <= h . r1 ) implies h is non-increasing )
assume A3: for r1, r2 being Real st r1 in dom h & r2 in dom h & r1 < r2 holds
h . r1 >= h . r2 ; :: thesis: h is non-increasing
let e1 be ExtReal; :: according to VALUED_0:def 16 :: thesis: for b₁ being object holds
( not e1 in dom h or not b₁ in dom h or not e1 <= b₁ or h . b₁ <= h . e1 )
let e2 be ExtReal; :: thesis: ( not e1 in dom h or not e2 in dom h or not e1 <= e2 or h . e2 <= h . e1 )
assume A4: ( e1 in dom h & e2 in dom h ) ; :: thesis: ( not e1 <= e2 or h . e2 <= h . e1 )
assume e1 <= e2 ; :: thesis: h . e2 <= h . e1
then ( e1 < e2 or e1 = e2 ) by XXREAL_0:1;
hence h . e2 <= h . e1 by A3, A4; :: thesis: verum