let X be set ; :: thesis: for F being PartFunc of X,ExtREAL holds

( F is nonnegative iff for n being Element of X holds 0. <= F . n )

let F be PartFunc of X,ExtREAL; :: thesis: ( F is nonnegative iff for n being Element of X holds 0. <= F . n )

let y be ExtReal; :: according to SUPINF_2:def 9,SUPINF_2:def 11 :: thesis: ( y in rng F implies 0. <= y )

assume y in rng F ; :: thesis: 0. <= y

then ex x being object st

( x in dom F & y = F . x ) by FUNCT_1:def 3;

hence 0. <= y by A2; :: thesis: verum

( F is nonnegative iff for n being Element of X holds 0. <= F . n )

let F be PartFunc of X,ExtREAL; :: thesis: ( F is nonnegative iff for n being Element of X holds 0. <= F . n )

hereby :: thesis: ( ( for n being Element of X holds 0. <= F . n ) implies F is nonnegative )

assume A2:
for n being Element of X holds 0. <= F . n
; :: thesis: F is nonnegative assume
F is nonnegative
; :: thesis: for n being Element of X holds 0. <= F . b_{2}

then A1: rng F is nonnegative ;

let n be Element of X; :: thesis: 0. <= F . b_{1}

end;then A1: rng F is nonnegative ;

let n be Element of X; :: thesis: 0. <= F . b

let y be ExtReal; :: according to SUPINF_2:def 9,SUPINF_2:def 11 :: thesis: ( y in rng F implies 0. <= y )

assume y in rng F ; :: thesis: 0. <= y

then ex x being object st

( x in dom F & y = F . x ) by FUNCT_1:def 3;

hence 0. <= y by A2; :: thesis: verum