let X be non empty set ; :: thesis: for f being Functional_Sequence of X,REAL

for x being Element of X st x in dom (sup f) holds

(sup f) . x = sup (rng (R_EAL (f # x)))

let f be Functional_Sequence of X,REAL; :: thesis: for x being Element of X st x in dom (sup f) holds

(sup f) . x = sup (rng (R_EAL (f # x)))

let x be Element of X; :: thesis: ( x in dom (sup f) implies (sup f) . x = sup (rng (R_EAL (f # x))) )

assume x in dom (sup f) ; :: thesis: (sup f) . x = sup (rng (R_EAL (f # x)))

then (sup f) . x = sup ((R_EAL f) # x) by MESFUNC8:def 4;

hence (sup f) . x = sup (rng (R_EAL (f # x))) by Th1; :: thesis: verum

for x being Element of X st x in dom (sup f) holds

(sup f) . x = sup (rng (R_EAL (f # x)))

let f be Functional_Sequence of X,REAL; :: thesis: for x being Element of X st x in dom (sup f) holds

(sup f) . x = sup (rng (R_EAL (f # x)))

let x be Element of X; :: thesis: ( x in dom (sup f) implies (sup f) . x = sup (rng (R_EAL (f # x))) )

assume x in dom (sup f) ; :: thesis: (sup f) . x = sup (rng (R_EAL (f # x)))

then (sup f) . x = sup ((R_EAL f) # x) by MESFUNC8:def 4;

hence (sup f) . x = sup (rng (R_EAL (f # x))) by Th1; :: thesis: verum