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

now

let x be Element of A; :: thesis: ((minfuncreal A) . f,((maxfuncreal A) . f,g)) . x = f . x
A1: x in dom (maxreal .: f,g) by Lm8;
A2: x in dom (minreal .: f,(maxreal .: f,g)) by Lm8;
thus ((minfuncreal A) . f,((maxfuncreal A) . f,g)) . x = ((minfuncreal A) . f,(maxreal .: f,g)) . x by Def5
.= (minreal .: f,(maxreal .: f,g)) . x by Def6
.= minreal . (f . x),((maxreal .: f,g) . x) by A2, FUNCOP_1:28
.= minreal . (f . x),(maxreal . (f . x),(g . x)) by A1, FUNCOP_1:28
.= f . x by Th13 ; :: thesis: verum

end;

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