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

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

let x be Element of A; :: thesis: ((minfuncreal A) . (((minfuncreal A) . (f,g)),h)) . x = ((minfuncreal A) . (f,((minfuncreal A) . (g,h)))) . x
A1: x in dom (minreal .: (f,g)) by Lm6;
A2: x in dom (minreal .: (g,h)) by Lm6;
A3: x in dom (minreal .: ((minreal .: (f,g)),h)) by Lm6;
A4: x in dom (minreal .: (f,(minreal .: (g,h)))) by Lm6;
thus ((minfuncreal A) . (((minfuncreal A) . (f,g)),h)) . x = ((minfuncreal A) . ((minreal .: (f,g)),h)) . x by Def5
.= (minreal .: ((minreal .: (f,g)),h)) . x by Def5
.= minreal . (((minreal .: (f,g)) . x),(h . x)) by A3, FUNCOP_1:22
.= minreal . ((minreal . ((f . x),(g . x))),(h . x)) by A1, FUNCOP_1:22
.= minreal . ((f . x),(minreal . ((g . x),(h . x)))) by Th4
.= minreal . ((f . x),((minreal .: (g,h)) . x)) by A2, FUNCOP_1:22
.= (minreal .: (f,(minreal .: (g,h)))) . x by A4, FUNCOP_1:22
.= ((minfuncreal A) . (f,(minreal .: (g,h)))) . x by Def5
.= ((minfuncreal A) . (f,((minfuncreal A) . (g,h)))) . x by Def5 ; :: thesis: verum

end;

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