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

now

let x be Element of A; :: thesis: ((minfuncreal A) . f,((maxfuncreal A) . g,h)) . x = ((maxfuncreal A) . ((minfuncreal A) . f,g),((minfuncreal A) . f,h)) . x
A1: x in dom (minreal .: f,g) by Lm8;
A2: x in dom (minreal .: f,h) by Lm8;
A3: x in dom (minreal .: f,(maxreal .: g,h)) by Lm8;
A4: x in dom (maxreal .: (minreal .: f,g),(minreal .: f,h)) by Lm8;
A5: x in dom (maxreal .: g,h) by Lm8;
thus ((minfuncreal A) . f,((maxfuncreal A) . g,h)) . x = ((minfuncreal A) . f,(maxreal .: g,h)) . x by Def5
.= (minreal .: f,(maxreal .: g,h)) . x by Def6
.= minreal . (f . x),((maxreal .: g,h) . x) by A3, FUNCOP_1:28
.= minreal . (f . x),(maxreal . (g . x),(h . x)) by A5, FUNCOP_1:28
.= maxreal . (minreal . (f . x),(g . x)),(minreal . (f . x),(h . x)) by Th14
.= maxreal . ((minreal .: f,g) . x),(minreal . (f . x),(h . x)) by A1, FUNCOP_1:28
.= maxreal . ((minreal .: f,g) . x),((minreal .: f,h) . x) by A2, FUNCOP_1:28
.= (maxreal .: (minreal .: f,g),(minreal .: f,h)) . x by A4, FUNCOP_1:28
.= ((maxfuncreal A) . (minreal .: f,g),(minreal .: f,h)) . x by Def5
.= ((maxfuncreal A) . ((minfuncreal A) . f,g),(minreal .: f,h)) . x by Def6
.= ((maxfuncreal A) . ((minfuncreal A) . f,g),((minfuncreal A) . f,h)) . x by Def6 ; :: thesis: verum

end;

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