let A be non empty set ; :: thesis: for f, g, h being Element of Funcs A,REAL holds (maxfuncreal A) . ((maxfuncreal A) . f,g),h = (maxfuncreal A) . f,((maxfuncreal A) . g,h)
let f, g, h be Element of Funcs A,REAL ; :: thesis: (maxfuncreal A) . ((maxfuncreal A) . f,g),h = (maxfuncreal A) . f,((maxfuncreal A) . g,h)
now let x be
Element of
A;
:: thesis: ((maxfuncreal A) . ((maxfuncreal A) . f,g),h) . x = ((maxfuncreal A) . f,((maxfuncreal A) . g,h)) . xA1:
x in dom (maxreal .: f,g)
by Lm8;
A2:
x in dom (maxreal .: g,h)
by Lm8;
A3:
x in dom (maxreal .: f,(maxreal .: g,h))
by Lm8;
A4:
x in dom (maxreal .: (maxreal .: f,g),h)
by Lm8;
thus ((maxfuncreal A) . ((maxfuncreal A) . f,g),h) . x =
((maxfuncreal A) . (maxreal .: f,g),h) . x
by Def5
.=
(maxreal .: (maxreal .: f,g),h) . x
by Def5
.=
maxreal . ((maxreal .: f,g) . x),
(h . x)
by A4, FUNCOP_1:28
.=
maxreal . (maxreal . (f . x),(g . x)),
(h . x)
by A1, FUNCOP_1:28
.=
maxreal . (f . x),
(maxreal . (g . x),(h . x))
by Th10
.=
maxreal . (f . x),
((maxreal .: g,h) . x)
by A2, FUNCOP_1:28
.=
(maxreal .: f,(maxreal .: g,h)) . x
by A3, FUNCOP_1:28
.=
((maxfuncreal A) . f,(maxreal .: g,h)) . x
by Def5
.=
((maxfuncreal A) . f,((maxfuncreal A) . g,h)) . x
by Def5
;
:: thesis: verum end;
hence
(maxfuncreal A) . ((maxfuncreal A) . f,g),h = (maxfuncreal A) . f,((maxfuncreal A) . g,h)
by FUNCT_2:113; :: thesis: verum