let C1, C2 be non empty set ; :: thesis: for f, g being RMembership_Func of C1,C2 holds converse (min f,g) = min (converse f),(converse g)
let f, g be RMembership_Func of C1,C2; :: thesis: converse (min f,g) = min (converse f),(converse g)
A1: ( dom (converse (min f,g)) = [:C2,C1:] & dom (min (converse f),(converse g)) = [:C2,C1:] ) by FUNCT_2:def 1;
for c being Element of [:C2,C1:] st c in [:C2,C1:] holds
(converse (min f,g)) . c = (min (converse f),(converse g)) . c

proof

let c be Element of [:C2,C1:]; :: thesis: ( c in [:C2,C1:] implies (converse (min f,g)) . c = (min (converse f),(converse g)) . c )
assume c in [:C2,C1:] ; :: thesis: (converse (min f,g)) . c = (min (converse f),(converse g)) . c
consider y, x being set such that
A2: ( y in C2 & x in C1 & c = [y,x] ) by ZFMISC_1:def 2;
A3: [x,y] in [:C1,C2:] by A2, ZFMISC_1:106;
(converse (min f,g)) . y,x = (min f,g) . x,y by A2, Def1
.= min (f . x,y),(g . x,y) by A3, FUZZY_1:def 4
.= min ((converse f) . y,x),(g . x,y) by A2, Def1
.= min ((converse f) . y,x),((converse g) . y,x) by A2, Def1 ;
hence (converse (min f,g)) . c = (min (converse f),(converse g)) . c by A2, FUZZY_1:def 4; :: thesis: verum

end;

hence converse (min f,g) = min (converse f),(converse g) by A1, PARTFUN1:34; :: thesis: verum