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 :: thesis: for x being Element of A holds ((maxfuncreal A) . (((maxfuncreal A) . (f,g)),h)) . x = ((maxfuncreal A) . (f,((maxfuncreal A) . (g,h)))) . x

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 Lm6;
A2: x in dom (maxreal .: (g,h)) by Lm6;
A3: x in dom (maxreal .: ((maxreal .: (f,g)),h)) by Lm6;
A4: x in dom (maxreal .: (f,(maxreal .: (g,h)))) by Lm6;
thus ((maxfuncreal A) . (((maxfuncreal A) . (f,g)),h)) . x = ((maxfuncreal A) . ((maxreal .: (f,g)),h)) . x by Def4
.= (maxreal .: ((maxreal .: (f,g)),h)) . x by Def4
.= maxreal . (((maxreal .: (f,g)) . x),(h . x)) by A3, FUNCOP_1:22
.= maxreal . ((maxreal . ((f . x),(g . x))),(h . x)) by A1, FUNCOP_1:22
.= maxreal . ((f . x),(maxreal . ((g . x),(h . x)))) by Th3
.= maxreal . ((f . x),((maxreal .: (g,h)) . x)) by A2, FUNCOP_1:22
.= (maxreal .: (f,(maxreal .: (g,h)))) . x by A4, FUNCOP_1:22
.= ((maxfuncreal A) . (f,(maxreal .: (g,h)))) . x by Def4
.= ((maxfuncreal A) . (f,((maxfuncreal A) . (g,h)))) . x by Def4 ; :: thesis: verum

end;

hence (maxfuncreal A) . (((maxfuncreal A) . (f,g)),h) = (maxfuncreal A) . (f,((maxfuncreal A) . (g,h))) by FUNCT_2:63; :: thesis: verum