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

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

let x be Element of A; :: thesis: ((minfuncreal A) . (f,g)) . x = ((minfuncreal A) . (g,f)) . x
A1: x in dom (minreal .: (f,g)) by Lm6;
A2: x in dom (minreal .: (g,f)) by Lm6;
thus ((minfuncreal A) . (f,g)) . x = (minreal .: (f,g)) . x by Def5
.= minreal . ((f . x),(g . x)) by A1, FUNCOP_1:22
.= minreal . ((g . x),(f . x)) by Th2
.= (minreal .: (g,f)) . x by A2, FUNCOP_1:22
.= ((minfuncreal A) . (g,f)) . x by Def5 ; :: thesis: verum

end;

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