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)) . x
A1: x in dom (maxreal .: f,g) by Lm8;
A2: x in dom (maxreal .: g,h) by Lm8;
A3: x in dom (maxreal .: (maxreal .: f,g),h) by Lm8;
A4: x in dom (maxreal .: f,(maxreal .: 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 A3, 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 A4, 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