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

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 Lm6;
A2: x in dom (minreal .: (f,h)) by Lm6;
A3: x in dom (minreal .: (f,(maxreal .: (g,h)))) by Lm6;
A4: x in dom (maxreal .: ((minreal .: (f,g)),(minreal .: (f,h)))) by Lm6;
A5: x in dom (maxreal .: (g,h)) by Lm6;
thus ((minfuncreal A) . (f,((maxfuncreal A) . (g,h)))) . x = ((minfuncreal A) . (f,(maxreal .: (g,h)))) . x by Def4
.= (minreal .: (f,(maxreal .: (g,h)))) . x by Def5
.= minreal . ((f . x),((maxreal .: (g,h)) . x)) by A3, FUNCOP_1:22
.= minreal . ((f . x),(maxreal . ((g . x),(h . x)))) by A5, FUNCOP_1:22
.= maxreal . ((minreal . ((f . x),(g . x))),(minreal . ((f . x),(h . x)))) by Th7
.= maxreal . (((minreal .: (f,g)) . x),(minreal . ((f . x),(h . x)))) by A1, FUNCOP_1:22
.= maxreal . (((minreal .: (f,g)) . x),((minreal .: (f,h)) . x)) by A2, FUNCOP_1:22
.= (maxreal .: ((minreal .: (f,g)),(minreal .: (f,h)))) . x by A4, FUNCOP_1:22
.= ((maxfuncreal A) . ((minreal .: (f,g)),(minreal .: (f,h)))) . x by Def4
.= ((maxfuncreal A) . (((minfuncreal A) . (f,g)),(minreal .: (f,h)))) . x by Def5
.= ((maxfuncreal A) . (((minfuncreal A) . (f,g)),((minfuncreal A) . (f,h)))) . x by Def5 ; :: thesis: verum

end;

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