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

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 Lm6;
A2: x in dom (minreal .: (f,(maxreal .: (f,g)))) by Lm6;
thus ((minfuncreal A) . (f,((maxfuncreal A) . (f,g)))) . x = ((minfuncreal A) . (f,(maxreal .: (f,g)))) . x by Def4
.= (minreal .: (f,(maxreal .: (f,g)))) . x by Def5
.= minreal . ((f . x),((maxreal .: (f,g)) . x)) by A2, FUNCOP_1:22
.= minreal . ((f . x),(maxreal . ((f . x),(g . x)))) by A1, FUNCOP_1:22
.= f . x by Th6 ; :: thesis: verum

end;

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