let A be non empty set ; :: thesis: for f, g being Element of Funcs A,REAL holds (maxfuncreal A) . f,g = (maxfuncreal A) . g,f
let f, g be Element of Funcs A,REAL ; :: thesis: (maxfuncreal A) . f,g = (maxfuncreal A) . g,f

now

let x be Element of A; :: thesis: ((maxfuncreal A) . f,g) . x = ((maxfuncreal A) . g,f) . x
A1: x in dom (maxreal .: f,g) by Lm8;
A2: x in dom (maxreal .: g,f) by Lm8;
thus ((maxfuncreal A) . f,g) . x = (maxreal .: f,g) . x by Def5
.= maxreal . (f . x),(g . x) by A1, FUNCOP_1:28
.= maxreal . (g . x),(f . x) by Th8
.= (maxreal .: g,f) . x by A2, FUNCOP_1:28
.= ((maxfuncreal A) . g,f) . x by Def5 ; :: thesis: verum

end;

hence (maxfuncreal A) . f,g = (maxfuncreal A) . g,f by FUNCT_2:113; :: thesis: verum