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

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 Lm8;
A2: x in dom (minreal .: g,f) by Lm8;
thus ((minfuncreal A) . f,g) . x = (minreal .: f,g) . x by Def6
.= minreal . (f . x),(g . x) by A1, FUNCOP_1:28
.= minreal . (g . x),(f . x) by Th9
.= (minreal .: g,f) . x by A2, FUNCOP_1:28
.= ((minfuncreal A) . g,f) . x by Def6 ; :: thesis: verum

end;

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