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

now

let x be Element of A; :: thesis: ((maxfuncreal A) . f,((minfuncreal A) . f,g)) . x = f . x
A1: x in dom (minreal .: f,g) by Lm8;
A2: x in dom (maxreal .: f,(minreal .: f,g)) by Lm8;
thus ((maxfuncreal A) . f,((minfuncreal A) . f,g)) . x = ((maxfuncreal A) . f,(minreal .: f,g)) . x by Def6
.= (maxreal .: f,(minreal .: f,g)) . x by Def5
.= maxreal . (f . x),((minreal .: f,g) . x) by A2, FUNCOP_1:28
.= maxreal . (f . x),(minreal . (f . x),(g . x)) by A1, FUNCOP_1:28
.= f . x by Th12 ; :: thesis: verum

end;

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